Feasibility Studies for Measuring
the Astrophysical S-Factor of the
Reaction 12C(α, γ)16O via
Electro-Disintegration at MAGIX
Dissertation
zur Erlangung des Grades
„Doktor der Naturwissenschaften“
am Fachbereich Physik, Mathematik und Informatik
der Johannes Gutenberg-Universität in Mainz
Stefan Lunkenheimer
Institut für Kernphysik
Johannes Gutenberg-Universität
Mainz, 22. November 2021
Autor
Stefan Lunkenheimer
MAGIX - Collaboration
Institut für Kernphysik
Johann-Joachim-Becher-Weg 45
Johannes Gutenberg-Universität
D-55128 Mainz
stelunke@uni-mainz.de
Tag der mündlichen Prüfung
3. März 2022
Abstract
The nucleosynthesis process 12C(α, γ)16O is of atmost importance for nuclear as-
trophysics. The S-factor of this reaction is a major parameter for determining
the abundance ratio of oxygen to carbon in stellar nucleosynthesis and plays a
crucial role in understanding the pre-supernova evolution. The current relative
error of more than 17% in the S-factor for energies in the Gamow-peak does not
allow a prediction of the abundance of higher mass elements produced in the pre-
supernova evolution of stars. Since the cross section of the reaction 12C(α, γ)16O
is low, a high-precision low-energy experiment with a high luminosity is required
to measure the S-factor. One such experiment that aims at measuring this key
and challenging quantity is MAGIX.
MAGIX is a versatile fixed-target experiment, which will be built at the new
electron accelerator MESA in Mainz, Germany. The accelerator will deliver po-
larized and unpolarized electron beams to MAGIX with currents up to 1mA at
105MeV. Using its internal gas jet target, MAGIX will reach a luminosity of
O(1035 cm−2 s−1). MAGIX will be equipped with two magnetic spectrometers,
that allow to measure the kinematic parameters of the scattered electrons with
a momentum resolution of δp < 10−4 and an angular resolution of ∆Θ < 0.05°.
p
To detect heavy charged particles, MAGIX will employ a silicon strip detector
array. This will allow studying processes with very low cross sections at small
momentum transfer within a rich physics program.
At MAGIX, a measurement is planned in which the S-factor will be determined by
investigating the electro-disintegration reaction 16O(e, e′α)12C. Electrons will be
scattered inelastically on oxygen and MAGIX will detect the scattered electrons
and the produced α-particles in coincidence. With this measurement, it will be
possible to determine the S-factor of the reaction 12C(α, γ)16O as a function of
the center of mass energy of the 12C-α system.
The simulations predict that MAGIX will be able to measure the S-factor for
center of mass energies above 1 MeV with a precision competitive to the existing
data. To improve these results, MAGIX will be operated in addition with a Zero
Degree Tagger to detect the scattered electrons in the range θe < 0.5°. Given the
promising simulation results, MAGIX expects to make important contributions
to the measurement of the S-factor of the reaction 12C(α, γ)16O down to 0.5 MeV,
exploring a new territory in nuclear astrophysics.
Keywords: MAGIX, MESA, Nuclear Astrophysics, Helium Burning, S-Factor,
12C(a,g)16O, Electro-Disintegration, 16O(e,e’a)12C, EPICS, Slow Control, Sim-
ulation
i
ii
Contents
Abstract i
Contents iii
1 Introduction 1
2 The Reaction 12C(α, γ)16O in the Helium Burning Stage in Stars 5
2.1 Overview of Stellar Nucleosynthesis . . . . . . . . . . . . . . . . . 5
2.1.1 Nuclear Reactions . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Cross Section of Stellar Nucleosynthesis . . . . . . . . . . . 7
2.1.3 Calculation of the Gamow Peak . . . . . . . . . . . . . . . 9
2.1.4 The Hydrogen Burning Stage . . . . . . . . . . . . . . . . 12
2.1.5 The Helium Burning Stage . . . . . . . . . . . . . . . . . . 16
2.2 The α-Capture Process of 12C in Detail . . . . . . . . . . . . . . . 20
2.3 Discussion of Measurement Techniques . . . . . . . . . . . . . . . 24
2.3.1 Direct Measurement Techniques . . . . . . . . . . . . . . . 24
2.3.2 Indirect Measurement Techniques . . . . . . . . . . . . . . 27
2.4 Results of Existing Measurements . . . . . . . . . . . . . . . . . . 29
2.5 Production Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Electro-Disintegration Cross Section of 16O 37
3.1 Kinematics of the Electro-Disintegration Reaction . . . . . . . . . 37
3.2 Differential Cross Section of the Electro-Disintegration Reaction
16O(e, e′α)12C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Calculation of the Matrix Element . . . . . . . . . . . . . . . . . 40
3.4 Definition of Spherical Unit Vectors . . . . . . . . . . . . . . . . . 41
3.5 Calculation of Generalized Rosenbluth Factors . . . . . . . . . . . 42
3.6 Calculating the Generalized Mott Cross Section . . . . . . . . . . 45
3.7 Combining Results to Differential Cross Section . . . . . . . . . . 45
3.8 Transformation to Center of Mass Frame . . . . . . . . . . . . . . 46
3.9 Approximation Using the Extreme Relativistic Limit . . . . . . . 48
3.10 Multipole Decomposition of the Response Functions . . . . . . . . 50
3.11 Representing the Electric Matrix Elements tEJ with the S-Factor
Components SEJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Accelerator and Experimental Setup 53
4.1 Mainz Energy-Recovering Superconducting Accelerator . . . . . . 53
4.1.1 The P2-Experiment . . . . . . . . . . . . . . . . . . . . . . 55
4.1.2 The DarkMESA Experiment . . . . . . . . . . . . . . . . . 56
4.2 Mainz Gas Injection Target Experiment . . . . . . . . . . . . . . 56
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Contents
4.2.1 Gas Jet Target . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2.2 STAR/PORT Spectrometer Setup . . . . . . . . . . . . . . 60
4.2.3 Silicon Strip Recoil Detector Array . . . . . . . . . . . . . 63
4.2.4 Zero Degree Tagger – FORE . . . . . . . . . . . . . . . . . 65
4.3 S-Factor Measurement at MAGIX . . . . . . . . . . . . . . . . . . 66
4.3.1 Discussion of Electro-Disintegration Experiment . . . . . . 67
4.3.2 Discussion of the Planned Extension Stages . . . . . . . . 68
5 Development of the MAGIX Slow Control System 71
5.1 Requirements for a Slow Control System . . . . . . . . . . . . . . 71
5.2 Experimental Physics and Industrial Control System . . . . . . . 72
5.2.1 Input-Output-Controller . . . . . . . . . . . . . . . . . . . 73
5.2.2 CA Client . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2.3 The Channel Access Communication Protocol . . . . . . . 77
5.3 Community Modules . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3.1 The Asyn Module . . . . . . . . . . . . . . . . . . . . . . . 78
5.3.2 The StreamDevice Module . . . . . . . . . . . . . . . . . . 79
5.3.3 The Autosave Module . . . . . . . . . . . . . . . . . . . . 80
5.3.4 The PyEpics Library . . . . . . . . . . . . . . . . . . . . . 80
5.3.5 The dev-gpio Module . . . . . . . . . . . . . . . . . . . . . 81
5.4 The Graphical User Interface - Control System Studio . . . . . . . 81
5.5 MAGIX Slow Control System . . . . . . . . . . . . . . . . . . . . 82
5.5.1 Installation Script . . . . . . . . . . . . . . . . . . . . . . . 82
5.5.2 Introduction to Environment Variables . . . . . . . . . . . 83
5.5.3 Execution via System Control Manager . . . . . . . . . . . 83
5.5.4 Defining UDEV Rules . . . . . . . . . . . . . . . . . . . . 85
5.5.5 Implementation of the Bash/ZSH Completion . . . . . . . 85
5.5.6 Integration of an Archiver (MAGIX executable) . . . . . . 86
5.5.7 Development of the MXFrontend (MAGIX module) . . . . 87
5.5.8 Expansion by the dbLoadMultiRecords Function . . . . . . 87
5.5.9 Development for Profibus Support (StreamDevice BUS API) 90
5.5.10 MXSlowControl Tutorial . . . . . . . . . . . . . . . . . . . 90
5.5.11 Development of Device Support for MAGIX Built Devices 90
5.6 Slow Control Development for the Silicon-Strip Detector Test Stand 93
5.6.1 Readout of the MPT200 Device via StreamDevice ttyUSB 93
5.6.2 Controlling the HMP4040 Device via StreamDevice TCP . 95
5.6.3 Driver Development to Control the SiDet board via RPi I2C 96
6 Simulation of the Electro-Disintegration Reaction 16O(e, e′α)12C103
6.1 Overview of Physics Simulations . . . . . . . . . . . . . . . . . . . 103
6.1.1 Principle of Monte Carlo Integration . . . . . . . . . . . . 104
6.1.2 General Idea of a Cross Section Simulation for a Scattering
Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.1.3 Initial State Particle Generator . . . . . . . . . . . . . . . 109
6.1.4 Final State Particle Generator . . . . . . . . . . . . . . . . 110
6.1.5 Physics Model Calculations . . . . . . . . . . . . . . . . . 110
6.1.6 Acceptance Simulation . . . . . . . . . . . . . . . . . . . . 111
6.1.7 Calculation of the Cross Section . . . . . . . . . . . . . . . 111
6.2 Implementation of the S-Factor Simulation . . . . . . . . . . . . . 112
6.2.1 Overview of the Initial State Generator . . . . . . . . . . . 112
6.2.2 Development of the Two-Body Final State Generator . . . 113
6.2.3 Development of the S-Factor Model Calculation . . . . . . 115
iv
Contents
6.2.4 Implementation of Acceptance Simulation . . . . . . . . . 116
6.2.5 Analysis: Calculation of Expected Count Rate . . . . . . . 118
6.2.6 Discussion of the Simulations Results . . . . . . . . . . . . 122
7 Summary and Outlook 127
A Response Functions 131
B Selected Data 135
C EPICS Support Files 141
D Supplemental Material – Simulation 157
List of Figures 161
List of Tables 163
List of Codes 165
Acronyms 167
Index 169
Bibliography 173
v
vi
CHAPTER 1
Introduction
Studying the interior of stars in laboratory experiments
During a clear night, one can see with modern telescopes millions of galaxies in the
sky, containing billions of stars. Some of these celestial objects are several billion
years old before their light reaches us. Since ever, people have wondered what
other solar systems and the planets to be found there might look like. In addition,
the question also arises whether there is life on other worlds and how this could
have evolved. To be able to provide answers to such questions, science depends on
observations of the universe and, as far as possible, on experiments in terrestrial
laboratories. This results in models, which can additionally be simulated. The
results of such simulations can then be compared with the observations. All these
endeavors improve our picture of the universe.
To understand the formation and evolution of solar systems, it is important to
understand the formation of elements in stellar interiors. This is called stellar
nucleosynthesis and describes the processes by which a star fuses hydrogen into
more massive elements such as helium, carbon, nitrogen, oxygen, etc. during its
lifetime. While our sun is still in the first stage, namely the hydrogen burning
stage, we can observe other stars that are already in later stages, such as the
helium or the carbon burning stage. To describe the evolution of a star, especially
the different burning stages, we need to understand the nuclear processes and
their reaction rates. Hereby, the cross section is an important quantity, which is
related to the probability of a certain reaction.
In the hydrogen burning stage, protons are fused to helium via different proton-
proton chains or carbon-nitrogen-oxygen cycles. The processes involved in this
phase are well known and have been investigated to such an extent that the timing
of this phase can be well determined. The situation is different with the subsequent
phase, the helium burning stage. Here, two reactions are crucial: the triple-α
process, in which a 12C is formed from three α-particles via a reaction chain, and
the fusion of carbon and helium to form oxygen 12C(α, γ)16O. These two processes
occur in parallel in the helium burning stage, but are in no sense to be understood
as a reaction chain. The second reaction, the α-capture of carbon, proceeds much
slower than the triple-α process, so that at the end of this burning stage only
1
Chapter 1 - Introduction
a small fraction of the carbon has been converted to oxygen. The description
of further processes and of the further evolution of a star depends sensitively
on the C/O abundance ratio. To estimate this ratio, one must determine the
cross section of the 12C(α, γ)16O reaction precisely. Therefore, the nucleosynthesis
process 12C(α, γ)16O is of utmost importance for nuclear astrophysics. For energies
available in the helium burning stage, the cross section of this reaction is low.
For this reason, this reaction can be studied in the laboratory only with large
efforts in experiments. An important parameter is the S-factor. This is the
unknown part of the cross section when the well-known parameters such as the
Coulomb penetration factor are removed. The S-factor of a fusion represents any
effect related to the nuclear structure. Despite efforts over the past 50 years to
determine the S-factor of this reaction, it has not yet been possible to determine it
with sufficient accuracy. Current relative errors are in the range of 17% to 28%.
In order to understand stellar evolution, and thus the further stages of burning
and the production of more massive elements, theory requires that the S-factor
must be determined in the low-energy range with an accuracy better than 10%.
In order to measure the cross section and thus the S-factor in the low energy range,
new low-energy experiments must be developed. These experiments must have a
high luminosity in order to be able to measure reactions with a low cross section
accurately. Only in this way it will be possible to further reduce the relative error
of the S-factor of the reaction 12C(α, γ)16O.
One such experiment planned for the coming years will be MAGIX at MESA.
MESA (Mainz Energy-Recovering Superconducting Accelerator) is an upcoming
high-intensity electron accelerator and is being built at the Institute of Nuclear
Physics of the Johannes Gutenberg University Mainz, Germany. MAGIX (Mainz
Gas Injection Target Experiment) will be located in the energy recovering branch
of MESA. The core element of MAGIX is the internal gas jet target. To measure
the kinematic parameters of the scattered electrons, MAGIX will use two mag-
netic spectrometers, each equipped with a GEM-based TPC in the focal plane. To
detect heavy charged particles, MAGIX will have a silicon strip detector array, lo-
cated within the scattering chamber. In a later stage, MAGIX will be extended to
include a beam-integrated tagging detector. MAGIX at MESA will allow studying
processes with very low cross sections at small momentum transfer. One experi-
ment planned is the determination of the cross section of the electro-disintegration
reaction 16O(e, e′α)12C. The scattered electron and the recoiling α-particle are
detected in coincidence to determine the kinematic parameters of this inelastic re-
action. Through theoretical calculations, the S-factor of the reaction 12C(α, γ)16O
can be estimated from the cross section of the electro-disintegration reaction of
oxygen.
The aim of this work was to perform a feasibility study using a simulation to an-
swer the question whether MAGIX can significantly contribute to the determina-
tion of the S-factor in the low energy range by measuring the electro-disintegration
reaction of oxygen.
Outline
At the beginning, the nucleosynthesis reaction 12C(α, γ)16O is discussed in detail.
For this purpose, the cross section and the S-factor of this reaction are introduced
and described with respect to the different oxygen levels. Furthermore, different
direct and indirect measurement methods are explained, which allow to determine
2
the S-factor or different contributions of the S-factor. Finally, the results of the
previous measurements performed by other research groups are discussed and an
overview of the impact of the results of these measurements on the production
factor is given. This is a quantity that describes the ratio of isotopes produced
during the pre-supernova evolution of massive stars.
While in Chapter 2 the direct reaction 12C(α, γ)16O was in the focus, in Chapter 3
the electro-disintegration reaction 16O(e, e′α)12C is discussed. The aim of this
chapter is to describe the theoretical basis of this reaction and to derive the cross
section. The theoretical assumptions and calculations of this chapter serve as a
basis for a simulation of the electro-disintegration measurement at MAGIX.
Subsequently, the experimental environment is presented in Chapter 4. All com-
ponents of MAGIX and MESA which are relevant for the electro-disintegration
measurement are described here. In order not to go beyond the scope of this
work, only the essential properties are described. At the end of this chapter, the
planned measurements of the electro-disintegration of oxygen are outlined.
MAGIX uses a slow control system based on EPICS. In Chapter 5, an overview
of the development of the MAGIX slow control system is given. Some parts of
MAGIX already exist and are tested in different laboratories. At the end of this
chapter, an example of setting up a test stand with EPICS is shown, namely the
test stand for silicon strip detectors.
After the theoretical studies have been completed and the experimental setup has
been described, a simulation is presented. Chapter 6 discusses the development
and results of the simulation of the electro-disintegration reaction of oxygen at
MAGIX at MESA. First, an overview of physics simulations is given. The principle
of Monte Carlo integration and random numbers is also introduced. Then, the
general idea of a cross section simulation for scattering experiments is presented.
Based on the previous discussions, it is then described in detail how the simulation
for the feasibility study is implemented in the MAGIX framework and how the
data will be analyzed. At the end of this section, the results will be discussed.
The final chapter of this thesis summarizes the results of this work. It also gives
a brief outlook on the S-factor measurement at MAGIX.
3
4
CHAPTER 2
The Reaction 12C(α, γ)16O in the
Helium Burning Stage in Stars
In this chapter we will take a closer look at the astrophysical S-factor of the
reaction 12C(α, γ)16O. The chapter starts with a brief introduction to nuclear
astrophysics and an overview of stellar evolution. Then the helium burning stage,
especially the associated S-factor of the α-capture process, will be discussed in de-
tail. Finally, we will assess the results of previous measurements used to determine
the S-factor and discuss why this is still challenging as of today.
2.1 Overview of Stellar Nucleosynthesis
The Big Bang nucleosynthesis describes the production of the first elements in
the early universe. After 3minutes, the main part of the baryon mass consisted of
hydrogen and helium, where the mass ratio of 4He to the total baryon mass was
∼ 25 %. This corresponds to a ratio by number of atoms of 4He/1H = 0.08 = 8 %
[Zyl+20; ISG13]. Traces of deuterium, tritium and lithium have also been pro-
duced, but their abundance compared to hydrogen is several orders of magnitude
lower. The ratio by number of atoms are D/1H = O(10−5), 3He/1H = O(10−5)
and, 7Li/1H = O(10−10) [Zyl+20, Chapter 24]. Therefore, the primordial elements
deuterium, tritium and lithium can be disregarded for the stellar evolution and
nucleosynthesis [Won98].
After the short time period of the Big Bang nucleosynthesis, the universe expanded
further and cooled down. However, it took another 100million years before the
first stars were formed [LB04]. The nuclear fusion between atoms within stars is
described by the stellar nucleosynthesis. It is a highly predictive theory that today
yields excellent agreement between calculations based upon it and the observed
abundances of the elements.
2.1.1 Nuclear Reactions
Nuclear reactions are processes in which a nuclear particle (nucleon or nuclei)
comes into close contact with another nuclear particle. This may produces one or
more nuclear particles that leave the interaction point. These produced particles
5
Chapter 2 - The Reaction 12C(α, γ)16O in the Helium Burning Stage in Stars
are usually different from the incident particle pair [BW91]. For a particle a that
scatters at a particle X the whole set of nuclear reactions can be written as
 X + a X∗ + a′
a+X −→

W +Nγ , (2.1)
Y + b
. . .
where a, b, . . . denotes the outgoing nuclear particle, W and Y, . . . denotes the
produced nucleus and Nγ denotes the emitted photons. With the exception of
the first reaction, these are inelastic nuclear reactions. The first reaction X(a, a)X
of Eq. (2.1) represents the elastic scattering process, where in the center of mass
system (cms) the incident particle a leaves the interaction with the same energy
and the target particle X stays in its initial state. The second reaction X(a, a′)X∗
represents the inelasitc scattering where the target particle X leaves the interac-
tion in an excited state X∗ and the outgoing particle a′ leaves with lower energy
than initial. The third reaction X(a,Nγ)W represents the radiative capture pro-
cess, where the two initial particles a andX stay together to form one final nucleus
W while one ore more photons γ are emitted. In the inelastic reactions, the strong
and electromagnetic forces are usually involved in order to exchange energy and
momentum. The weak nuclear force can also take place in such a reaction, for
example to convert neutrons into protons via the β− processes [BW91; Won98].
All inelastic reactions can be subdivided based on the quantum states of the final
particles. The set (α′, α′′) denotes the states of the initial particles and the sets
{(β′, β′′), (γ′, γ′′), . . . } the one of the final particles. For a fixed inital channel
(α′, α′′) we can rewrite Eq. (2.1) as

 Xα′ + aα′′ X + a′β′ β′′
aα′′ +Xα′ −→  Wω + nγ . (2.2) Yγ′ + bγ′′. . .
Due to the conservation laws of energy, angular momentum, and parity, the out-
going quantum states of each set {(β′, β′′), . . . } are restricted. All possible sets
with fixed quantum state are called reaction channels. Each reaction channel is
abbreviated as β ≡ (β′, β′′), for example [BW91].
Furthermore we have to distinquish between direct and compound reactions. In
case of a direct reaction the two initial particles interact once with each other in a
short time of the order of 10−22 s. If the two initial particles „fuse“ together for a
long time typically of the order of 10−14 s the reaction is called compound reaction.
In case of the compound reaction an intermediate state, a so called compound
nucleus N , is formed. Before any re-emission occurs, the energy of the incident
particle is quickly shared through the strong force between all constituents. An
important feature is that in a compound reaction the decay of the compound
nucleus is independent of the formation of this nucleus. So, one compound nucleus
6
2.1 Overview of Stellar Nucleosynthesis
can have several final reaction channels [Won98; BW91] X + a′ β′ β′′
Nα −→

Wω + nγ
. (2.3)
Yγ′ + bγ′′
. . .
2.1.2 Cross Section of Stellar Nucleosynthesis
The cross section is an important quantity to define the reaction rate of reactions.
It is a measure of the probability with which a given reaction occurs for given
initial and final particles. Assuming a scattering process in which an incident
particle beam is scattered at a layer of independent scattering centers, the total
cross section can be defined as [Pov+15; BW91]:
:= number of reactions per unit timeσtot .number of incident particles per unit time× scattering centres per unit area
The dimension of the cross section is area, which is usually given in cm2 or barn
which is defined as
1 barn = 1 b = 10−24 cm2.
The total cross section σtot is a function that depends only on the cms energy E
of a given reaction. The total cross section can be written as an integral of the
differential cross section over the total solid angle [Fey04]
∫
σ(E) = dσ
4π dΩ
dΩ .
The differential cross section describes the cross section for scattering into a small
solid angle element dΩ. It depends on the cms energy E and the cms scattering
angle θ, which is defined as the angle between the incident and final particles in
the cms (see Fig. 2.1). In order to describe the dependence on the angle, one can
use the multipole expansion to represent the differential cross section in terms of
the Legendre polynomials Pk(cos(θ)) as [Ros53; Fey04]
dσ ∑
dΩ = akPk(cos(θ)), (2.4)k
where the factors ak ≡ ak(E) depend only on the cms energy E, with a0 =
σ
4 .π
b
a θ X
Y
Figure 2.1: Schematic view of a scattering process in the cms. The scattering
angle θ is defined as the angle between the incident and final particles.
7
Chapter 2 - The Reaction 12C(α, γ)16O in the Helium Burning Stage in Stars
For an initial channel α, the total cross section of a scattering process can be
separated into the elastic cross section σel and the combined cross section of all
inelastic processes σinel, also called the reaction cross section [Pov+15; BW91]:
σtot(α) = σel(α) + σinel(α).
Moreover, the inelastic cross section can be divided into all possible reaction
channels β according to (2.2) [Pov+∑15; BW91]:
σinel(α) = σ(α, β) + σcap(α), (2.5)
β 6=α
where σ(α, β) is the inelastic scattering cross section with the initial channel α
and the resulting channel β and σcap is the cross section of the radiative capture
process. Here, the cross section σ(α, β) is defined as [BW91]
( ) = number of emissions into channel β per unit time per scattering centerσ α, β .
number of incident particles in channel α per unit area per unit time
In stellar nucleosynthesis studies, we are interested in the inelastic cross section
for a given reaction with specific initial and final particles. Since the constituents
interact through the strong force, we have to distinguish between two cases, the
resonant and the nonresonant.
For the inital channel α and the final channel β the cross section close to a
resonance with the energy Er can be written with the Breit-Wigner formula as
[Bur+57; BW91; Pov+15]
( ) = 2 (2J + 1) ΓαΓβσ(α,β) E πλ̄ (2 ·Ja + 1)(2Jb + 1) ( )2 + Γ2 4
, (2.6)
E − Er tot/
where λ̄ ≡ λ̄(E) is the reduced de Broglie wavelength in the cms, Ja and Jb are
the spin for the initial particles, J the spin for the final nucleus, Γα and Γβ the
partial widths for the inital and final channels, and Γtot the total width. The total
width is the sum of the partial widths of all possible final channels. This cross
section is often written as
σ(E) = 2 · Γtotπλ̄ ωγ ( , (2.7)E − Er)2 + Γ2tot/4
with the resonance strength [deB+17]:
= (2J + 1) · ΓαΓβωγ (2 .Ja + 1)(2Jb + 1) Γtot
If the energy falls between resonances the cross section is given as [Bur+57]
σ(E) = πλ̄2(2J + 1)(π2 ± 8)ΓαΓβ , (2.8)
D2
with the average level spacing D. If the two nearest resonances interfere construc-
tive the factor (π2 +8) is used, otherwise if they interfere deconstructive the factor
(π2 − 8) is used.
We can now describe the cross section for stellar nucleosynthesis reactions. In
case of stellar nucleosynthesis the temperature is around a few GK and hence
the most probable energy is in the order of a few keV. It is therefore suitable to
8
2.1 Overview of Stellar Nucleosynthesis
use the non-relativistic approximation to calculate the kinetic energy E = 12µv
2,
with the reduced mass µ and the relative velocity v of the incident particles. In
combination with Eqs. (2.5) to (2.8) it follows that the cross section in this low
energy region is proportional to λ̄2, whereby [Fey04]
λ̄2 ∝ 1 1
v2
∝ ,
E
with the cms energy E.
Moreover, the probability for a nucleus with the proton number Z1 tunneling
through the Coulomb barrier of a nucleus with proton number Z2 with relative
velocity v ≡ v(E) is described by the Coulomb penetration factor
−2πZ
e 1
Z2αemc
v .
It is common to define the Sommerfeld number
≡ ( ) := Z1Z2αemcη η E , (2.9)
v
with the fine structure constant αem and the speed of light c. Since only charged
nuclear particles are involved, the nucleosynthesis cross section contains the Cou-
lomb penetration factor
e−2πη.
In total we have
σ(E) = 1 e−2πη · S(E),
E
where the so called S-factor is defined as the contribution of the nucleosynthesis
cross section without the known dependencies [Fey04]
S(E) := σ(E) · E · e2πη. (2.10)
It is usally given in the units keVb or MeVb. The specific S-factor of a compound
reaction represents any effect pertaining to the nuclear structure [Won98].
2.1.3 Calculation of the Gamow Peak
To describe the stellar nucleosynthesis it is crucial to know the energy range in
which the reactions take place. On the one hand, the Maxwell-Boltzmann distri-
bution describes the probability to find a particle in a certain energy state. For a
gas at the temperature T the Maxwell-Boltzmann distribution p(v) as function of
the relative velocity v between two (arbitrar)y particles can be written as [Won98]3/2 −µv2
p(v) dv = 4 µπ e 2 2kT2 v dv , (2.11)πkT
with the reduced mass µ and the Boltzmann constant k (see Table B.8). The
reduced mass is calculated with
µ = M1M2 ,
M1 +M2
whereMi with i = 1, 2 denotes the mass of the two initial particles. The maximum
point of p(v) is given by 12µv
2
max = kT . In Fig. 2.2 a set of Maxwell-Boltzmann
distributions for different elements for different burning stages in stars can be
9
Chapter 2 - The Reaction 12C(α, γ)16O in the Helium Burning Stage in Stars
1H @T
2 9
= 0.02
D @T9 = 0.023He @T = 0.02
4 9He @T9 = 0.0212C @T = 0.02
14 9
1× 10−6 N @T16 9 = 0.02O @T9 = 0.02
0
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008
Velocity v [c]
(a) Maxwell-Boltzmann distribution for different elements involved in the hydrogen
burning stage T9 = 0.02. The velocity always refers to the relative velocity be-
tween the respective element and hydrogen.
1× 10−6 4He @T = 0.2
8 9Be @T
12 9
= 0.2
C @T9 = 0.216O @T
4 9
= 0.2
He @T
12 9
= 0.8
C @T9 = 0.84He @T = 2.0
12 9C @T9 = 2.016O @T9 = 2.0
0
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
Velocity v [c]
(b) Maxwell-Boltzmann distribution for different elements involved in the helium burn-
ing stage T9 = 0.2, the carbon burning stage T9 = 0.8 and the oxygen burning stage
T9 = 2.0 [Won98]. The velocity always refers to the relative velocity between the
respective element and 4He.
Figure 2.2: Maxwell-Boltzmann distribution for specific elements at the tem-
peratures for√specific burning stages in stars. T9 describes the temperaturein units of GK. The maximum point of each distribution is at the relative
velocity v̂ = 2kTµ for the specific reduced mass µ (see Table B.5).
10
Probability p(v) [s/m]
Probability p(v) [s/m]
2.1 Overview of Stellar Nucleosynthesis
found. As mentioned above it is suitable to calculate the kinetic energy using the
non-relativistic approximation E = 1 2max 2µvmax = kT .
On the other hand, the probability for an element to tunnel through the Coulomb
barrier of another element is given by the Coulomb penetration factor. For the
energy range of a stellar fusion, we can express the Coulomb penetration factor
c(E) as a function of the cm{s energy as √ } { }
( ) := exp −2 µ = exp −√bc E πZ1Z2αemc 2 , (2.12)E E
√
with b = πZ1Z2αemc 2µ.
The mean reaction rate λ per unit volume per unit time in a gas mixture with
the particles densities n1 and n0 is defined as the integral over the product of rel-
ative velocity, the cross section and the Maxwell-Boltzmann distribution [Won98;
CHY92; Bur+57]: ∫ ∞
λ := n1n0 〈σv〉 = n1n0 σ(E) · v · p(v) dv . (2.13)
0
The substitution of E = 1µv2∫2 with dE = µv dv yields∞
λ = 1n1n0 ∫ σ(E) · p(E) dµ 04 ∞ ( (
E )
= π S E)
3/2
e−√
b µ e− E · 2E√ n1n0 E { kT dµ 0 E ∫ 2πkT µ }
E
= 8 1
∞ b
3 2n1n0 S(E) exp −√ −
E dE . (2.14)
µπ (kT ) / 0 E kT
The integrand can be interpreted as the reaction rate distribution, called Gamow
peak. If we assume that there are no resonances involed in the region the reaction
takes place, the nuclear cross section factor S(E) can assumed to be essentially
smooth. To calculate the maximum of the reaction rate distribution it is sufficient
to calculate the maximum of the exponential function exp (−f(x)) with
f( ) := √bE + E .
E kT
The exponential function has a maximum if and only if the function f has a
minimum. Therefore we calcu∣∣latedf
d ∣∣∣ = 1 − b =! 0.E kT 2 3/2E=E EG G
By solving this equation for the energ(y we ge1 )
t
2/3
EG = 2bkT . (2.15)
In the following chapters we will refer to the energy EG as Gamow peak. A
schematic view of the Maxwell-Boltzmann distribution, the Coulomb penetration
factor and the Gamow peak can be found in Fig. 2.3.
The full width of the Gamow peak is defined as the difference between the two
energy values at which the Gamow function is equal to the maximum value divided
11
Chapter 2 - The Reaction 12C(α, γ)16O in the Helium Burning Stage in Stars
by the Euler’s number e. To calculate the width of this distribution the Gamow
peak is approximat{ed by a norm}al distribution{with(the mean)va}lue EG:
b E 2exp −√ − ≈ Amax · exp −
1 E − EG
2 , (2.16)E kT σ
where Amax defines the maximum value and σ the standard dev√iation of the normal
distribution. So, the width can be calculated as ∆EG = 2 2 · σ. To get the
width ∆EG we must compare Eq. (2.16) and the first two derivatives at the point
E = EG. If w{e evaluate Eq.}(2.16) for E = EG we get 3/2 { }
exp −√b E
 
− G = exp− 2E√G − EG = exp −3EGEG kT{ ( kT)}EG kT kT
=! Amax · exp −4
EG − EG
∆ = Amax.EG { }
Hence, the maximum value is given as Amax = exp −3EG . In order to simplifykt
the calculation of the derivatives, let u(s make u)se of the following formula
dn ∑n dkg
d exp{g(x)} = d · exp{g(x)}, (2.17)xn kk=1 x
for each differentiable function g(x).
Taking a closer look at Eq. (2.16), it is immediately noticeable that at E = EG
the exponential functions result in the value 1. So, for all derivatives we only must
equate the sum of the derivatives of the(exponen)t function. We will now define
( ) := 4 E − E
2
G
g E
∣ ∆
.
EG ∣
By definition it is clear, that df ∣ dg ∣d ∣ = 0 and d ∣ = 0, because bothE E=E EG E=EG
functions have a local minimum point at E = EG.
Now, let’s calculate the second derivatives of the exponent functions f and g
according to the energy E evaluated at EG:
d2f ∣∣∣
d 2 ∣∣
2 { } [ ]
= d b E 3 −5/2 3 1
E dE2
√ + = bE = ,
E kT 4 E=EG 2 kTEG
d2g ∣∣∣E=EG { ( )E}=EG2 2
d 2 ∣∣ = dd 2 4 E − EG 8E = E ∆ = 2 .EE E GG E= ∆EEG G
Finally, we can express the width of the√Gamow peak as [Won98; Fey04]
∆ = 4 EGkTEG 3 . (2.18)
2.1.4 The Hydrogen Burning Stage
This subsection will give a brief overview of the important reaction chains of the
hydrogen burning stage by pointing out the most important points of the books
Introductory Nuclear Physics [Won98, Chapter 10], The Physics of Stars [Phi99,
Chapter 4], and Evolution of Stars and Stellar Populations [SC05, Chapter 5].
12
2.1 Overview of Stellar Nucleosynthesis
Maxwell-Boltzmann Coulom( b PenetrationDistribution Factor √ )
∝ exp(−E/kT ) ∝ exp −b/ E
Gamow-Peak
∆EG
kT EG Energy
Figure 2.3: Sketch of the Gamow peak, Maxwell-Boltzmann distribution and
Coulomb penetration factor. For better illustration, the Coulomb penetration
factor as well as the Gamow peak are drawn several orders of magnitude larger.
The first stage of stellar evolution is the hydrogen burning stage. In this stage
the driving factor is hydrogen. The temperature raises up to T9 = 0.02, where T9
describes the temperature in units of GK [Won98]. From the Maxwell-Boltzmann
distribution Eq. (2.11) follows that the kinetic energy is kT ≈ 2 keV. Due to the
high density and temperature within the core of a star a plasma is formed. Since
the energy is significantly higher than the binding energy of the electrons, the
plasma consists of protons and free electrons. The net result of this burning stage
is the fusion of four protons into a 4He nucleus [Won98]. However, such a direct
four-body reaction is extremely unlikely and the fusion process is seperated into
the Proton-Proton chains (PP chains) and the Carbon-Nitrogen-Oxygen cycle
(CNO cycle). While the PP chain is involved in the fusion all the time, it is
necessary for the CNO cycle that carbon, nitrogen or oxygen atoms are already
present, as they act as catalysts [SC05].
The Proton-Proton Chains
Let’s now have a closer look at the Proton-Proton chains (PP chains). A shematic
view can be found in Fig. 2.4. The first reaction of this chain is the production
of deuteron through the colission of two protons
p+ p→ d+ e+ + νe. (2.19)
During the short timescale of this direct reaction, a proton is converted by the
weak interaction into a neutron, a positron and an electron neutrino. Since a
bound system with two protons is not possible and the weak interactions is involed,
the probability of this reaction is very low even at stellar densities. In fact, the
cross section of this reaction is so low that no experimental measurments are
possible on earth [SC05; Phi99]. For a star with central density ρ = 105 kg m−3,
temperature T9 = 0.015 and hydrogen mass fraction X = 0.5, the reaction rate
in the center of the star is λ = 5× 1013 s−1 m−1. This means that a proton in
13
Probability
Chapter 2 - The Reaction 12C(α, γ)16O in the Helium Burning Stage in Stars
p+ p→ d+ e+ + νe
p+ d→ 3He + γ 3He + 4He→ 7Be + γ
3He + 3He→ 4He + p+ p 7Be + e− → 7Li + ν 7 8e Be + p→ B + γ
7Li + p→ 4He + 4He 8B→ 8Be + e+ + νe
8Be→ 4He + 4He
PPI Chain PPII Chain PPIII Chain
Figure 2.4: Schematic view of the Proton-Proton I-III chains. As a net result
each of the three chains combines four protons to one 4He.
the center has to stay on average about 9× 109 years before it is converted to
deuterium by the reaction (2.19). This illustrates how slow this process proceeds
and that this reaction is the bottleneck in hydrogen burning and sets the time
scale for this phase in a star. All further reactions in the PP chains proceeds much
faster. [Phi99].
The second reaction in the PP chain is the fusion of a proton and a deuteron to
3He
p+ d→ 3He + γ. (2.20)
Since this is an electromagnetic reaction the cross section is in the order of O(1018)
greater than the cross section of the weak reaction Eq. (2.19). Thus, the deuteron
is almost immediately converted to 3He by the reaction Eq. (2.20) [Phi99].
At this point multiple possible reactions can be processed to fuse 3He to higher
mass elements. One possibility is that an 3He nucleus may tunnel through the
Coulomb barrier of another one to form 4He
3He + 3He→ 4He + p+ p. (2.21)
The reaction chain of the three reactions Eqs. (2.19) to (2.21) is known as the
PPI chain. On the left-hand side of this reaction chain a total of six protons
are involved while on the right-hand side one 4He, two protons, two positrons,
two electron neutrinos, and two γ-rays are produced [Won98]. Using the Einstein
relation E = mc2 the energy released by this reaction chain can be calculated
easily as 26.731 MeV [SC05].
The presence of 4He opens up another possible reaction for the fusion of 3He
3He + 4He→ 7Be + γ. (2.22)
Due to the high electron density within the core of a star, the newly formed
beryllium nuclei can be converted to lithium by the weak interaction
7Be + e− → 7Li + νe. (2.23)
14
2.1 Overview of Stellar Nucleosynthesis
At the end of the PPII chain, the fusion of lithium and a proton results in two
helium nuclei [Won98].
7Li + p→ 4He + 4He. (2.24)
Even in the PPII chain, where 4He act as catalyst, a total of four protons are
converted into one 4He nucleus [Phi99].
The PPIII chain is also connected to the production of 7Be from helium Eq. (2.22).
But instead of an electron capture, the beryllium can fuse with a proton to form
boron [Won98]
7Be + p→ 8B + γ. (2.25)
While 8B is unstable, it decays via e+-emission into 8Be
8B→ 8Be + e+ + νe. (2.26)
As 8Be isn’t stable, it decays through α-emission
8Be→ 4He + 4He. (2.27)
The PPIII chain also produces one 4He out of four protons in total.
The Carbon-Nitrogen-Oxygen Cycle
As discussed previously, a further possibility to convert protons into 4He is the
Carbon-Nitrogen-Oxygen cycle (CNO cycle). The CNO cycle mainly consists of
two independent cycles, the CN and the NO cycle. To start the reactions within
these cycles the presence of the elements C, N, and O are essential as they act as
catalysts [SC05]. Both cycles consists of three main reactions. The first reaction
(p, γ) is the fusion of a nucleus with a proton to another nucleus. The second
reaction is the β+-decay of a nucleus, and the third one is the α-emission reaction.
The reactions involved in the CN cycle are [Won98; SC05]
p+ 12C → 13N + γ
13N → 13C + e+ + νe
p+ 13C → 14N + γ
p+ 14N → 15O + γ
15O → 15N + e+ + νe
p+ 15N → 12C + α.
While carbon and nitrogen act as catalysts in the CN cycle, the catalysts in the
NO cycle are nitrogen and oxygen. The reactions involved in the NO-cycle are
[Won98; SC05]:
p+ 14N → 15O + γ
15O → 15N + e+ + νe
p+ 15N → 16O + γ
p+ 16O → 17F + γ
17F → 17O + e+ + νe
p+ 17O → 14N + α.
15
Chapter 2 - The Reaction 12C(α, γ)16O in the Helium Burning Stage in Stars
The result of both cycles are similar to the result of the PP-Chains: four protons
are converted into one 4He, two positrons, two electron neutrinos, and three γ-
particles. The two reactions are connected via the nucleus 14N, 15O and 15N.
Figure 2.5 shows a schematic view of these two cycles. There are other side cycles
that we will not discuss in this work. A more detailed overview can be found in
Introductory Nuclear Physics [Won98], for example.
Although the CNO cycle was predicted by Bethe and Weizsäcker already in the
1930s, the first experimental evidence was obtained in 2020 by the Boroxino ex-
periment [Ago+20].
12 (p, α)C 15
(p, γ)
N 16O
(p, γ) β+ (p, γ)
13N CN 15O NO 17Cycle Cycle F
β+ (p, γ) β+
13 (p, γ)C 14
(p, α)
N 17O
.
Figure 2.5: Schematic view of the CNO cylce which consists of the two in-
depentend cycles, the CN and the NO cycle. The elements Carbon, Nitrogen
and Oxygen act as catalysts and their presence is essential for the two cycles.
In total four protons are converted into one 4He nucleus, two positrons, two
electron neutrinos, and three γ-particles.[Ago+20; SC05; Phi99; Won98]
2.1.5 The Helium Burning Stage
This subsection will give a short overview of the helium burning stage by sum-
marizing the key messages of the books Introductory Nuclear Physics [Won98,
Chapter 10-5], The Physics of Stars [Phi99, Chapter 4], and Evolution of Stars
and Stellar Populations [SC05, Chapter 6].
When most of the hydrogen within the core of a star is fused to helium, the
reaction rate of the PP chains and the CNO cycle decreases. Since the temperature
is not sufficient to drive fusion reactions to produce higher mass elements, the
hydrostatic equilibrium cannot be maintained, whereby gravity begins to compress
the core of the star. This increases the central temperature up to T9 = 0.2, at
which the helium burning takes place [Sal52]. The remaining hydrogen continues
the process of forming helium in the outer / cooler layers of the star, but with a
lower reaction rate due to the lower density [Won98].
The helium burning stage consists of two main reactions. The first fundamental
one is the fusion of three 4He nuclei to 12C. This is called the triple-α reaction.
The second reaction is the production of higher mass nuclei via the α-capture
reaction. The triple-α process and the most important α-capture reaction, the
production of 16O, are shown schematically in Fig. 2.6.
16
2.1 Overview of Stellar Nucleosynthesis
Figure 2.6: Reaction scheme of the two most important reactions of the helium
burning stage. The triple-α reaction is shown in the green rectangle, where
two helium particles fuse to beryllium, which fuses to carbon through another
α-capture. The α-capture reaction of 12C ist shown in the red rectangle.
Triple-α Reaction
In the triple-α reaction, three 4He nuclei fuse to one 12C. A simultaneous collision
of three helium atoms is very unlikely, so the triple-α process usually takes place
in two steps [FF14; Phi99; Sal52]:
4He + 4He 8Be,
8Be + 4He 12C∗(7.654 MeV)→ 12C + {2γ or (e+ + e−)}.
The ground state of 8Be is 91.84 keV above the 2α threshold (see Fig. 2.7). The
production of 8Be starts at a temperature at around T9 = 0.12, because then the
Gamow energy EG = 92.3 keV with a width of ∆EG = 72.1 keV is sufficient to
drive this fusion reaction (see also Table B.6) [FF14; Phi99; Sal52].
Since the ground state of 8Be is above the 2α threshold, it is unstable. It decays
with a width of Γ = 5.57 eV (see also Table B.1) back to 2α. However, the ground
state of 8Be is a metastable state, which means that its lifetime is several orders of
magnitude longer than that of other states which decay by the strong interaction
(∼ 10−22 s) [BW91]. In fact, its half-life of
τ1/2 =
~ ln(2)
Γ = 8.19× 10
−17 s
is a factor ∼ 105 larger than that of the nearest excited states (see Table B.1).
This leads to the fact that in the core of a star a state of dynamic equilibrium
of 4He and 8Be is formed. The number density ratio can be calculated using the
formula for the population of 8Be nuclei in a gas of 4He nuclei at temperature T
[Phi99] [ ] 3 [ ]
= 2π~M
2 2
8Be 8
n8Be exp −
(M Be − 2Mα)c · (n 24He) ,
MαkT kT
with n4He, n8Be the number density of 4He, 8Be and Mα,M8Be the nuclear masses
of 4He, 8Be. For a central density of ρ = 108 kg m−3, the 4He number density is
n = ρ4He = 1.5× 1034 m−3 for example. At T9 = 0.12 the 8Be number density isMα
n8Be = 4.37× 1025 m−3 giving a ratio of about 8Be : 4He = 1 : 3 · 108. However, at
T9 = 0.2 the 8Be number density increases to n8Be = 7.16× 1026 m−3, resulting in
17
Chapter 2 - The Reaction 12C(α, γ)16O in the Helium Burning Stage in Stars
a ratio of about 8Be : 4He = 1 : 2 · 107. Although the 8Be number density is very
low, its presence opens up the next stage of the triple-α process [Phi99].
The carbon production is strongly enhanced by the core structure of carbon.
The excited state 0+ at 7.654 MeV of carbon, known as the Hoyle state, lies
287 keV above the threshold of 8Be + α and furthermore in the Gamow window
(see Fig. 2.7). The value of the Gamow peak for T9 = 0.2 is EG = 232 keV with
the width ∆EG = 146 keV (see also Table B.6). This boosts the capture process
by a factor of 10 – 100million and as a result, most of the carbon in the core of a
star is produced through the Hoyle state [FF14].
The Hoyle state seems to form a spatial structure, where three α-cluster are
arranged nearly linearly [Epe+11; Hjo11]. It has a finite lifetime with a half-life
of τ1/2 ≈ 5× 10−17 s and decays in more than 99 % of all cases into three α (see
also Table B.2). Only in one of 2403.85 cases the Hoyle state transits to the
ground state of 12C via either two-gamma-decay (0+ → 2+ → 0+) or e+e−-pair
production (0+ → 0+) [FF14]. The formation of 12C via the Hoyle state is crucial
for nuclear astrophysics and the production of higher elements [FF14; Won98].
α-Capture Reaction
In the helium burning stage, higher mass elements are formed by the α-capture
reaction. The temperature for this reaction is around T9 = 0.2 [SC05; Won98].
The following α-capture reactions take place in the helium burning stage [SC05]:
12C + α → 16O + γ,
16O + α → 20Ne + γ,
20Ne + α → 24Mg + γ,
24Mg + α → 28Si + γ.
All of these reactions are compound reactions. Oxygen is mainly produced through
the two subthreshold states 1− at 7.116MeV and 2+ at 6.917MeV (see Fig. 2.7).
The cross section for this reaction at the energy of the Gamow peak EG ≈ 300 keV
is in the order of 10−17 barn. We will discuss this reaction in more detail in the
next section.
With an increasing proton number Z the Coulomb penetration factor is decreas-
ing exponentially. For example, at the cms energy Ecms = 300 keV the Coulomb
penetration factor for the reaction 16O(α, γ)20N is about O(10−23), which is a
factor ∼ 106 smaller than for the reaction of 12C(α, γ)16O (see Table B.7). Fur-
thermore, at T9 = 0.2 the Gamow window of the reaction 16O(α, γ)20N is close
to the 2− state at 4.947 MeV of 20Ne (see Fig. 2.7). Since the initial particles of
the reactions are in the ground state 0+ this state is of unnatural parity. The
term of (un)natural parity is further discussed in Section 2.2. As a consequence,
the production of 20Ne is highly suppressed in the helium burning stage [Won98,
p. 379]. Since the Coulomb penetration factor decreases exponentially with in-
creasing charge number the production of magnesium and silicon is also highly
suppressed in the helium burning stage (see also Table B.7).
Eventually, the triple-α process and the α-capture of Carbon play an important
role in the helium burning stage, while at T9 = 0.2 other fusion reactions can be
neglected since they proceed much slower. Therefore a mixture of 12C, 16O and
only traces of 20Ne are formed [SC05]. One of the most important reactions in the
stellar evolution is the reaction of 12C(α, γ)16O, since the production of elements
18
2.1 Overview of Stellar Nucleosynthesis
+
Triple- 9.870 2α
E[MeV] Jπ 9.641 3−
3.03 2+
+
8 7.654 0Be + α (7.367 MeV) EG
4He + α (−0.092 MeV) 0 0+ E
8 GBe 4.439 2+
α-Capture 9.844 2+
9.585 1−
8.871 2−
0 0+ +α
12
12C + α (7.161 MeV) 7.116 1− E
C
G
−0.045MeV
−0.245MeV 6.917 2+
6.129 3−
6.049 0+ 6.725 0+
5.788 1−
5.621 3−
16O + Eα (4.73 MeV) G
4.967 2−
4.248 4+
1.634 2+
0 0+
16O
0 0+ +α
20Ne
Figure 2.7: Level scheme of the nuclei involed in the helium burning stage.
The blue line shows the invariant mass of the initial particles in relation to the
level scheme of the produced particle. The red bar shows the Gamow window
of each reaction at T9 = 0.2. In the triple-α reaction the carbon nucleus
is produced through the 0+ resonance at E = 7.654 MeV. In the further α-
capture reactions the most important one is the α-capture of carbon. The
oxygen is formed via the two subthreshold resonances with Ecms1− = −45 keV
and Ecms2+ = −245 keV (green lines). In the helium burning stage with T9 =
0.2 the α-capture reaction of oxygen can be neglected. On the one hand the
Coulomb penetration factor is several orders of magnitude lower, and on the
other hand the reaction is suppressed by the unnatural parity of the 2− state
at E = 4.967 MeV close to the Gamow window. All data can be found in the
Tables B.1 to B.6.
in further burning stages is sensitive to the C/O abundance ratio. This is often
referred as the holy grail in nuclear astrophysics. Many stellar models use the
S-factor of this reaction as an input parameter. So, the study of the reaction
12C(α, γ)16O plays a crucial role in understanding the evolution of stars.
19
Chapter 2 - The Reaction 12C(α, γ)16O in the Helium Burning Stage in Stars
2.2 The α-Capture Process of 12C in Detail
In this section we will discuss the astrophysical reaction 12C(α, γ)16O in more
detail. Hereby we will focus on the theoretical part, a discussion on the results of
previous measurements can be found in Section 2.3. For an even more detailed
description please refer to the papers [DB74; Bru01; BSS14; deB+17; FDM19].
All S-factor values used in this thesis were determined with the AZURE2 software
[Ud15; Azu+10] using the configuration parameters given in the supplementary
material from [deB+17]. These parameters represent the results of analysis of
previous data from [deB+17]. AZURE2 can be used to calculate differential and
angle-integrated cross sections for resonant nuclear reactions relevant to nuclear
12 +
E[MeV] Jπ C(0 )+α (14.815 MeV)
4α (14.437 MeV)
13.869 4+
13.664 1+
13.259 3
13.020 2+
12.440 1
15
12.049 0+ N + p (12.127 MeV)
11.600 3 12C(2+)+α (11.601 MeV)
11.520 2+
11.096 4+
10.356 4+
9.844 2+
9.585 1
8.871 2
EG 12
7.116 1 C + α (7.161 MeV)
−0.045MeV
6.917 2+ −0.245MeV
S-Factor [keVb]
6.129 3
6.049 0+
0 0+
16O
Figure 2.8: Level diagram of the 16O nucleus. The corresponding S-factor for
the reaction 12C(α, γ)16O is shown on the left, calculated with the AZURE2
software [Ud15] configured with the parameters of the supplemental material
provided by Ref. [deB+17]. The S-factor strongly depends on the nuclear
structure of 16O. Different thresholds are shown in blue on the right side.
The two levels a few keV below the carbon-α threshold play an important role
as they have the greatest impact in the area of the Gamow peak, shown in red.
20
103
102
1010 1 2 3 4 5 6
Center of Mass Energy [MeV]
2.2 The α-Capture Process of 12C in Detail
astrophysics. The open-source code AZURE2 is based on an R-matrix code that
can also be used for data fitting [Ud15; Azu+10].
The S-factor, Eq. (2.10), is characterized by the resonances, the non-resonant
contribution and the interference between these components [deB+17]. The con-
tribution strength of all these parts strongly depends on the nuclear structure
of 16O. Figure 2.8 shows the level structure of oxygen and the corresponding S-
factor of the reaction 12C(α, γ)16O. Since 16O is an even-even nucleus the ground
state has quantum numbers Jπ = 0+, with total spin J and parity π. Below the
threshold of 12C+α at 7.161MeV, four bound states are located at 0+(6.049 MeV),
3−(6.129 MeV), 2+(6.917 MeV), and 1−(7.116 MeV). The two states 1− and 2+ are
only a few keV below the threshold and contribute the most to the cross section in
the area of the Gamow peak. Therefore they play a decisive role in this reaction
[deB+17].
Oxygen is produced from the fusion of 12C and α, which are also 0+ nuclei. Due
to the conservation of parity only states with a total spin of J = l and parity
of π = (−1)l can arise, where l denotes the relative orbital angular momen-
tum. In addition, only electric transitions to the ground state can occur as de-
scribed by the γ-ray decay selection rule. Furthermore, transitions from 0+ to
0+ states are prohibited [deB+17]. This can also be seen in Fig. 2.8, so for ex-
ample the excited state 0+(12.049 MeV) does not contribute to the S-factor. At
this point it should be mentioned that nearly 100% of all excited states decay
to the ground state via γ-decay (see Table B.3). Exceptions are, for example the
states 4+(10.356 MeV) and 4+(11.096 MeV), which can only decay into the states
2+(6.917 MeV), 3−(6.129 MeV). Any transition that passes through one or more
intermediate states to the ground state is called a cascade transition. This ex-
plains that the largest contribution to the S-factor in the low-energy range near
the Gamow peak is made by the ground state transition, see Fig. 2.9.
Since 16O is a mirror-symmetric nucleus, the E1-decay into the ground state is
strictly isospin forbidden [BSS14]. The Coulomb interaction breaks the isospin
symmetry, so that such a transition is possible, but the strength is strongly reduced
[deB+17]. As a result, the contribution of the electric multipolar components E1
and E2 to the total S-factor for low energy levels is of similiar importance, see
Fig. 2.10. To summarize we can state that the oxygen is produced through the
two subthreshold states 1− at 7.116MeV and 2+ at 6.917MeV. The E1 and E2
contribution dominate the total S-factor in the area of the Gamow peak while
the contribution of the cascade transitions is small compared to the ground state
transition in this region.
Following the previous discussion, we can limit ourselves to the E1 and E2 multi-
polarities when calculating the differential cross section Eq. (2.4) for low energies.
The differential cross section in the center of mass frame of carbon-α as a function
of [the cms]energy E and photon angle θγ is therefore given as [DB74; deB+17]
dσ
dΩcms (E, θγ) =
σE1
4 [Q0P0(cos θγ)−Q2P2(cos θγ)] (2.28)γ ( πα,γ) [ ]
+ σE2 54 Q0P0(cos θγ) + 7Q2P2(cos θγ)−
12
7 Q4P4(cos θγ)π
+ √6
√
cos (φ12(E))
σE1σE2
4 [Q1P1(cos θγ)−Q3P3(cos θγ)],5 π
where Pn(cos θγ) are the Legendre polynomials, Qn the geometric attenuation
21
Chapter 2 - The Reaction 12C(α, γ)16O in the Helium Burning Stage in Stars
103
102
101
Total
Ground State Transition
100 Cascade Transitions
0 EG 1 2 3 4 5 6
Ecms [MeV]
Figure 2.9: The total S-factor of the reaction 12C(α, γ)16O compared to the
ground state transition and the sum of all cascade transitions. The curves are
calculated with the AZURE2 software [Ud15] configured with the parameters
of the supplemental material provided by Ref. [deB+17]. The ground state
transition dominates the total S-factor near the Gamow peak, while the sum of
all cascade transitions makes a small contribution in this region. The Gamow
peak is indicated by the blue dashed line and the Gamow width by the gray bar
on the left hand side.
factors (see [Ros53]), and φ12 the Coulomb phase shift difference between the E1
and E2 transition matrix elements. The phase shift difference can be written as
[DB74; deB+17] ( ( ))
cos(φ12) = cos δα1 − δα2 − arctan
η
2 , (2.29)
where δα1/2 are the elastic scattering phase shifts for angular momentum l = 1
and l = 2, respectiveley, and η the Sommerfeld number, see Eq. (2.9). For cms
energies Ecms ≤ 1.5 MeV we can approximate the phase shift to [Bru01]
cos( 2φ12) = √ 2 . (2.30)η + 4
The Coulomb phase shift and the approximated one can be seen in Fig. 2.11.
22
S-Factor S(E) [keV b]
2.2 The α-Capture Process of 12C in Detail
103
102
101
100
10−1 E1
E2
Cascades
10−2
0 EG 1 2 3 4 5 6
Ecms [MeV]
Figure 2.10: The E1 and E2 ground state contribution of the S-factor of the
reaction 12C(α, γ)16O compared to the contribution of cascade transitions. The
curves are calculated with the AZURE2 software [Ud15] configured with the
parameters of the supplemental material provided by Ref. [deB+17]. The
contributions of E1 and E2 to the S-factor near the Gamow peak are in the
same order of magnitude. The Gamow peak is indicated by the dashed blue
line and the Gamow width by the gray bar on the left hand side.
AZ(URE21 )Fit
2 · 2 + 4 −1/2η
1987 Plaga
0.8
0.6
0.4
0.2
0
0 0.5 1 1.5 2 2.5
Ecms[MeV]
Figure 2.11: For cms energies Ecms ≤ 1.5 MeV the Coulomb phase shift
cos(φ12) can be approximated using Eq. (2.30). The black line represents the
results from the R-Matrix fit using the AZURE2 software [Ud15] configured
with the results of Ref. [deB+17] provided as supplemental material. The red
line represents the approximation of the Coulomb phase shift using Eq. (2.30).
The blue dots represent the values of the Coulomb phase shift determined from
the elastic α-scattering 12C(α, α)12C data from [Pla+87].
23
cos(φ12) S-Factor S(E) [keV b]
Chapter 2 - The Reaction 12C(α, γ)16O in the Helium Burning Stage in Stars
2.3 Discussion of Measurement Techniques
Experiments to determine the total S-factor of the reaction 12C(α, γ)16O have
been carried out since 1955. The cross section at the temperature of the helium
burning stage (EG ≈ 300 keV) is around O(10−17 b). So, measurements are re-
ally challenging. There is currently no experiment which has measured the total
cross section below Ecms = 1 MeV (see Fig. 2.14). For low energies the Coulomb
penetration factor dominates the cross section and thus it decreases exponentially.
In the low energy range, the cross section falls by 6 orders of magnitude from
σ ∼ 4× 10−11 b at E = 1 MeV to σ ∼ 2× 10−17cms b at Ecms = 0.3 MeV (see the
black curve in Fig. 2.13). In order to determine the S-factor at the Gamow peak,
it is necessary to extrapolate the results of the measurements at higher energies to
lower energies. But the potential extrapolation errors must be taken into account.
It is therefore essential to determine the S-factor as precisely as possible in order to
reduce these errors. As described in Section 2.2, the cross section is influenced by
the nuclear structure of 16O. In particular, the cross section contains resonances
and interferences between these resonances. In order to extrapolate it to lower
energies, a method taking all of these components into account must be used.
These calculations require a high computational effort, as there are 2n−1 possible
interferences for n involved states. An appropriate method for such a calculation
is the R-matrix method, which is explained in detail in [deB+17]. To be able to
use this method, each multipolarity must be separated. Therefore it is important
to measure the angular distribution of the emitted photons during the experiment.
The E1 and E2 components of the ground state transition can then be determined
using Eq. (2.28). As described in Section 2.2, these two multipolarities dominate
the cross section for small energies. At higher energies, contributions from cascade
transitions and further multipolarities also become relevant. At this point we also
have to mention, that the determination of the E2 contribution in the range 1
to 3MeV is challenging since the total cross section is dominated by the broad
1−(9.585 MeV) resonance (see Figs. 2.8 and 2.10).
In this section we will discuss some experiments in more detail. At this point we
do not provide a complete list of all previous experiments, a detailed description
can be found in [deB+17]. In the following we will focus on some of the most
important experiments. We can roughly divide the experiments into direct and
indirect methods. In the direct method, the reaction 12C(α, γ)16O is studied.
Whereas the indirect methods derives the cross section from the measurements of
other nuclear reactions based on theoretical models.
2.3.1 Direct Measurement Techniques
The basic idea for the direct measurement of the reaction 12C(α, γ)16O is to carry
out α-carbon scattering experiments and to detect the recoil particles. The S-
factor can be determined from this measurement data. There are two possible
settings: the first uses an α-beam that scatters on a solid carbon target, the
second is employing the inverse kinematics where a 12C beam scatters on a 4He
gas target. Both methods have their individual advantages and disadvantages,
which we will discuss below. With direct measurements, it is almost impossible
to measure in the Gamow energy range. The cross section must therefore be
measured at higher energies and extrapolated to the Gamow peak [Fey04].
24
2.3 Discussion of Measurement Techniques
α-Particle Scattering on Carbon
The first method to measure the S-factor of the reaction 12C(α, γ)16O is the α-
particle scattering experiment at a solid carbon target. Hereby the emitted photon
will be detected to determine the angular distribution, in order to separate the
E1 and E2 contributions. In most cases a NaI-detector array is used to detect
the emitted photons. A major problem is contamination of the carbon target.
Traces of 13C atoms in the target material generate a high neutron background
through the reaction 13C(α, n)16O. It should be mentioned that the cross section
of reaction 12C(α, γ)16O at the peak of the 1−(9.585 MeV) resonance is more than
6 orders of magnitude smaller than the cross section of the reaction 13C(α, n)16O.
This is discussed in detail in [deB+17]. This large background made it difficult
to determine the S-factor for a long time. Higher purerity targets and better
methods of extracting the background were developed that ultimately made the
determination of cascade transitions possible.
The analysis of the data is challenging. As already mentioned, the E1 and E2
contributions are in the same order of magnitude. Equation (2.28) shows how the
two components contribute to the differential cross section. Figure 2.12 shows the
contribution factors of E1 and E2 in relation to this formula. It is immediately
noticeable that the E1 component can be isolated at 90°, where it is maximal,
while the E2 component vanishs. Thus, E1 can be determined by measuring the
cross section at θγ = 90°. It then app(lie)s( )
σE1 = 4
2 dσ
π 3 dΩ .
θγ=90°
Unfortunately, E2 cannot be isolated. Therefore a different method must be used
to determine E2. Intuitively, one would fit the data with Eq. (2.28). However, in
practise it shows, that this is quite difficult, since the correlation between σE1 and
σE2 is large due to the interference term
√
σE1σE2. It turns out that it is better to
u[se a mo]dified formula to fit the data [Kun+02; DB74]:
dσ σ E1
dΩcms (E, θγ) = 4 Q0P0(cos θγ)−Q2P2(cos θγ) (2.31)γ ( ) πα,γ [ ]
+ σE2 5 12Q0P0(cos θγ) + 7Q2P2(cos θγ)− 7 Q4P4(cos θγ)σE1 √ 
+ √6 cos (φ12(E))
σE2 · [Q1P1(cos θγ)−Q3P3(cos θγ)] .5 σE1 
In this formula, the parameters are given by the amplitude σE1, the ratio σE2σE1
and the phase difference φ12. There are two options to fit the data. The first
option is to perform a three-parameter fit. Since the phase difference can also
be determined via the elastic scattering 12C(α, α)12C, this parameter can also be
used as an input for the fit. The second option is to use a two-parameter fit
including a fixed-phase. To do this, one determines σE1 and the ratio σE2 and canσE1
hence determine σE2. However, due to the error propagation, we expect a lower
accuracy for E2 [deB+17; Ass+06].
There are several measurements planned to determine the cross section of the
reaction 12C(α, γ)16O. Two of these experiments are for example the LUNA-MV
experiment at the Gran Sasso laboratory and an experiment at the Felsenkeller
25
Chapter 2 - The Reaction 12C(α, γ)16O in the Helium Burning Stage in Stars
2
E1
1.8 E2
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0 20 40 60 80 100 120 140 160 180
θ [deg]
Figure 2.12: Theoretical angular contribution of E1 and E2 for the ground
state transition of 12C(α, γ)16O. The contribution is calcluted in relation
to Eq. (2.28), so that E1 represents P0(cos θ) − P2(cos θ) and E2 represents
P (cos θ)+ 50 7P2(cos θ)−
12
7 P4(cos θ). At 90°, the E1 contribution can be isolated
from the data, since it is at its maximum there, while the E2 contribution van-
ishs. Since there is no angle at which E2 can be perfectly isolated, determining
the E2 contribution requires more effort [deB+17].
laboratory in Dresden [Gug14; Bem+18]. One challenge in the direct measure-
ment is to detect the low-energy photons with sufficient resolution. This will be
improved further through the continuous development of the detectors. Due to
the low reaction rate near the Gamow peak, a major challenge is to separate the
12C(α, γ)16O signals from the much larger signals arising from the cosmic back-
ground. In this case, a location in the underground is the one solution to extremely
reduce cosmic rays [Gug14; Bem+18]. For example, the LUNA experiment is
comissioned 1400m deep under the Gran Sasso mountain. The muon component
of the cosmic background is therefore reduced by a factor of 106 [Gug14].
Measurements in Inverse Kinematics
Another method to suppress the background of 13C is to measure the reaction
in inverse kinematics. For this purpose, a high-intensity 12C beam scatters at an
4He gas target. With this method, the S-factor can be determined in two different
ways. As in the α-scattering case, the emitted photons can be detected in order to
separate the E1 and E2 contributions. However, the expansion of the gas target
and the forward kinematics must be taken into account for the analysis. This
makes the analysis more difficult compared to the measurement of α-scattering
on carbon. The second way to determine the S-factor is to detect the recoiling
oxygen. In the laboratory system it is emitted in a small window in the forward
direction and must be detected with high efficiency. The challenge at this point
is to separate the recoiling oxygen from the scattered carbon nuclei. For one 16O
nucleus around 1016 - 1018 12C nuclei are detected. The advantage is that with the
detection of the recoiling oxygen, the total S-factor can be determined, also includ-
26
Contribution Factor
2.3 Discussion of Measurement Techniques
ing the cascade transitions. Since it is impossible to perform this measurement
for low cms energies, this technique cannot be used to determine the S-factor near
the Gamow peak. An extrapolation without knowledge of the γ distribution is
not possible. Even though this method only provides useful results at higher cms
energies (E & 1 MeV), it still can be helpful. Other experiments, for example, can
compare their results from the separations with those of the total S-factor before
extrapolating them to the Gamow peak [deB+17; Fey04].
2.3.2 Indirect Measurement Techniques
There is a large number of indirect techniques to determine the S-factor of the
reaction 12C(α, γ)16O. The basic idea of all indirect techniques is to measure
related nuclear reactions with cross sections that are much larger than the one for
12C(α, γ)16O. Theoretical models are used to calculate the S-factor of the direct
reaction afterwards. For example the β-delayed α-emission of 16N can be used
to extract the α partial width of the E1 ground state transition [Fey04]. A full
discussion of all these methods can be found in [deB+17]. We will only focus on
two indirect methods in this section.
The Photo-Disintegration Measurement
The photo-disintegration reaction 16O(γ, α)12C is one of the most promising in-
verse techniques. The idea is to break up oxygen nuclei with a γ-ray beam. The
recoiled α-particles are then detected to determine the angular distribution. For
a reaction A(a, b)B, the reciprocity theorem connects the cross sections of in-
terchanged initial and final states. In the center of mass frame, the reciprocity
theorem can be expressed as [May02, chapter 7.3]:
σ(a, b) = (2Ib + 1)(2I
2
B + 1) |p~b |
( ) (2 + 1)(2 + 1) · 2 , (2.32)σ b, a Ia IA |p~a|
where I denotes the angular momentum for the particles involved and p~ the three-
momenta in the center of mass frame for the particles in the initial and final state.
In the area of the Gamow peak the cross section of the time-reversed reaction
16O(γ, α)12C is by a factor of ∼ 50 higher than the cross section of the direct
reaction 12C(α, γ)16O (see Fig. 2.13). This is an advantage of this method. There
have been two different experimental approaches so far. One of them uses a bubble
chamber as target, which also acts as a detector for the α-particle at the same time
[HFP19; HFP18; DiG+15; Uga+13]. Another experiment uses a CO2-operated
optical readout time projection chamber in which the outgoing α-particles are
detected. Corresponding experiments are currently under development and the
first measurements have already been made [Gai12]. For this purpose, an intensive
nearly monoenergetic γ-ray beam was used at the HIγS facility. This delivers
around 5× 107 γ/s to 3× 109 γ/s for energies in the range of 1MeV to 20MeV
[Zim+13; Wel+09].
Currently, the largest limiting factor of the photo-disintegration experiment is the
energy spread of the photons, which is about 200 keV for the experiments men-
tioned above [deB+17]. Since the initial oxygen is in the ground state, this method
can only be used to determine the cross section of the ground state transition.
But this method can still be used to make important contributions to the deter-
mination of the total S-factor as the cross section in the area of the Gamow peak
is dominated by the ground state transition of E1 and E2. The advantage of this
method is, on the one hand, the higher effective cross section, which allows lower
27
Chapter 2 - The Reaction 12C(α, γ)16O in the Helium Burning Stage in Stars
10−2
10−4
10−6
10−8
10−10
10−12
10−14
16O(γ, α)12C
10−16 12C(α, γ)16O
0.3 0.5 1 2 4 8
Ecms [MeV]
Figure 2.13: The cross section of the time-reversed reaction 16O(γ, α)12C can
be calculated using the reciprocity theorem Eq. (2.32). In the area of the
Gamow peak the cross section of the time-reversed reaction is by a factor
of ∼50 higher than the cross section of the direct reaction 12C(α, γ)16O.
energy regions to be measured with sufficient statistics. On the other hand, this
setup allows a very precise measurement of the angular distribution of the outgo-
ing α-particles, which means that the E1 and E2 components can be separated
very well. Experiments at the Extreme Light Infrastructure - Nuclear Physics
facility (ELI-NP) are planned, which aim to achieve a higher γ-flux in addition to
a better energy resolution [Gal+18; Mat+18; Bal17; Tes+16; Ton+05]. To sum it
up, photo-disintegration measurement is a promising method to perform precise
low energy measurements of the E1 and E2 components of the S-factor [deB+17].
The Electro-Disintegration Measurement
The measurement of the electro-disintegration reaction 16O(e, e′α)12C is an exten-
sion of the photo-disintegration measurement 16O(γ, α)12C [FDM19]. Electrons
are scattered inelastically on oxygen. The four-momentum of the virtual photons
can be determined by measuring the four-momentum transfer in the scattering
process. Therefore, the scattered electrons and the produced α-particles have to
be detected in coincidence. Finally, the S-factor as a function of the cms energy of
the 12C-α system can be determined from the electro-disintegration cross section.
The cross sec[tion in the cente]r of mass frame can be written as [FDM19]:
dσ M cmsαM12C pα σMott
dE ′d Ω d Ωcms = 8 (2.33)e α ( ′ ) π3W (~c)3e,e α
· (ṽLRL + ṽTRT + ṽTLRTL + ṽTTRTT ),
where W is the cms energy and pcmsα the three-momentum of the α-particle in
the center of mass frame. The functions ṽK are factors that depend on the elec-
tron kinematics and RK are the nuclear response functions: the longitudinal and
transverse nuclear electromagnetic current components and the two interference
28
σ [barn]
2.4 Results of Existing Measurements
terms (longitudinal L and transversal T with respect to the direction of the virtual
photon). Hereby the transverse nuclear electromagnetic current component RT is
connected to the differential cross section Eq. (2.28) of the direct reaction in the
limit Q2 → 0 via [FDM19]: [ ]
~c · pcms
R = α ·W dσT 2α · ωcms · 4π·MαM12C dΩcms
, (2.34)
γ (α,γ)
with the electron scattering energy transfer ω. A full derivation of the cross section
can be found in Chapter 3. For further information please refer to the paper by
Friščić, Donnelly, and Milner [FDM19].
There are some advantages and disadvantages compared to the photo-disintegra-
tion measurement. A major disadvantage is that the longitudinal and interference
terms are currently unknown. These terms must also be determined experimen-
tally and taken into account in the extrapolation for small energies. Another
disadvantage is that contamination by 17O, 18O or 14N of the gas target creates
a background that has to be extracted using suitable filters. An advantage is
that the background, induced by cosmic rays, is greatly reduced by the coinci-
dence measurement of the electron and the recoiled α-particle. Furthermore, the
virtual photon flux is significantly higher and can be controlled by the scatter-
ing angle and beam energy. Finally, the energy resolution of the photon can be
determined with high precision by detecting the scattered electron in a detector
with high resolution. Nevertheless, this method is an excellent extension of the
indirect methods that will also be carried out in the near future in the upcoming
experiment MAGIX in Mainz [FDM19].
2.4 Results of Existing Measurements
In this section, we will briefly review selected previous results on the measurement
of the S-factor of the 12C(α, γ)16O. All results can be found in Table 2.1. A de-
tailed discussion of all the previous results can be found in [deB+17]. In Fig. 2.14
the measured data at low energies separated by the electric transitions E1 and E2
are presented. As discussed in the previous section, these contributions dominate
the S-factor in the Gamow peak region, so we will focus on them in this section.
When taking a closer look at the data, we immediately notice that no data exists
in the energy range below 0.9MeV. Thus, there is a large gap where the data must
be extrapolated to the Gamow peak. The figure also includes an R-matrix fit and
extrapolation performed with the AZURE2 software [Ud15] configured with the
results of Ref. [deB+17] provided as supplemental material. The R-matrix fit
correlates well with the previous experimental data. Even though the S-factor
increases towards the Gamow peak, we must keep in mind that the cross section
for small energies is dominated by the Coulomb penetration factor (see Section 2.2
and Fig. 2.13). The cross sections drops by more than 6 orders of magnitude in
the range 1MeV to 0.3MeV. This makes the determination of the S-factor in the
range below 1MeV so challenging.
Several methods can be used to analyse the data and calculate the S-factor
S(300 keV). The R-matrix method (RM) is an excellent tool to analyze the data,
taking into account the different states, the interference between them as well
as the different transitions. However, this method is computationaly very inten-
sive. In many previous works, a specialized or simplified formalism was used, such
as the three-level R-matrix method (TLRM) or the five-level R-matrix method
29
Chapter 2 - The Reaction 12C(α, γ)16O in the Helium Burning Stage in Stars
(FLRM), in which only the first three or five states were considered for the various
electric ground state transitions [deB+17; Sch+12].
Before discussing the results of historical works, we must mention that the previous
analyses indicate that the interference between the two states 1−(7.116 MeV) and
1−(9.585 MeV) is constructive. To rule out destructive interference between these
two states with high confidence, data with high precision and accuracy below
Ecms = 1.3 MeV are required [Sch+12]. Most of the previous works assumed
constructive interference, unless we mention otherwise.
Dyer and Barnes 1974 extrapolated the S-factor for the E1 ground state transition
using two different methods. In addition to the TLRM, they used the hybrid R-
matrix method (HRM), in which data from elastic scattering 12C(α, α)12C was
also included. The results are [DB74]
STLRM(300 keV) = 140+140E1 −120 keV b SHRME1 (300 keV) = 80+50−40 keV b.
Also Redder et al. (1987) used the TLRM and the HRM to extrapolate the E1
ground state transition. They also determined the S-factor for the E2 ground
state transition using the FLRM and the HRM. In addition, they calculated
the contribution of the two cascade transitions via the states 2+(6.917 MeV) and
1−(7.116 MeV) using TLRM and HRM. The results are [Red+87]
STLRME1 (300 keV) = 200+270−110 keV b SHRME1 (300 keV) = 140+120−80 keV b
SFLRME2 (300 keV) = 96+24−30 keV b SHRME2 (300 keV) = 80± 25 keV b
STLRM6.92 (300 keV) = 7± 2 keV b SHRM6.92 (300 keV) = 6± 3 keV b
S7.12(300 keV) = 1.3± 0.5 keV b
When comparing the results of Dyer and Barnes; Redder et al. for the different
methods, the results for E2 are comparable, but the results differ significantly for
the E1 transition. This model dependency is one problem in analyzing the data
to determine the S-factor.
To improve the results Ouellet et al. (1996) performed a complete analysis of the
data sets from the earlier work of [DB74; Red+87; Kre+88], including the phase
shifts from elastic scattering experiments of the reaction 12C(α, α)12C [Pla+87]
and the β-delayed α spectrum from the decay of 16N [Azu+94]. For fitting and
extrapolating, they used the TLRM and an alternative method to the R-matrix,
the K-matrix method (KM). The results for the E1 ground state transition are
[Oue+96]
STLRME1 (300 keV) = 73.3± 13.2 keV b SKME1 (300 keV) = 84.8± 15.7 keV b
They combined all their results to calculate the E1 and E2 ground state transition
and the total S-factor
SE1(300 keV) = 79± 16 keV b SE2(300 keV) = 36± 6 keV b
S(300 keV) = 120± 40 keV b.
In addition, they used both methods to calculate the E1 ground state transition
for destructive interference between the two 1−(7.116 MeV) and 1−(9.585 MeV)
states
SE1(300 keV) = 2.4± 1 keV b.
This clearly shows the effect of the potentially destructive interference on the cross
section.
30
2.4 Results of Existing Measurements
Roters et al. (1999) used the FLRM to calculate the S factor of the E1 ground
state transition incorporating earlier data. The analysis results in [Rot+99]
SFLRME1 (300 keV) = 79± 21 keV b.
This result is comparable with the KM and TLRM results of Ouellet et al. as well
as the HRM results of Dyer and Barnes; Redder et al.
Gialanella et al. (2001) performed a full R-matrix analysis of their data, which
included the previous data, the phase shifts from elastic scattering, and the α
sprectrum data of the β-delayed 16N decay. For the first time they used a Monte
Carlo (MC) simulation to calculate the systematic uncertainties of the differences
between the different E1 data sets [Gia+01; deB+17]. Another result of theirs was
that destructive interference between the two lowest E1 ground state transitions
cannot be rigorously ruled out. The results are [Gia+01]
Sconstr.E1 (300 keV) = 82± 16 keV b Sdestr.E1 (300 keV) = 2.4± 1 keV b.
Kunz et al. (2001) studied the angular distribution of the S-factor measurements
in the low energy region 0.95MeV to 2.8MeV in detail. Using the TLRM for E1
and the FLRM for E2 in combination with the phase shifts from elastic scattering
and the α sprectrum data of the β-delayed 16N decay they determined the E1
and E2 ground state transition, the sum of all cascade transitions and the total
S-factor [Kun+01]
STLRME1 (300 keV) = 76± 20 keV b SFLRME2 (300 keV) = 85± 30 keV b
Scasc(300 keV) = 4± 4 keV b S(300 keV) = 165± 50 keV b.
The results of the data taken by Fey (2004) as part of his doctoral dissertation are
published in [Ham+05]. They used the same technique as Kunz et al. to determine
the E1 and E2 ground state transition and the total S-factor [Ham+05; Fey04]
STLRME1 (300 keV) = 77± 17 keV b SFLRME2 (300 keV) = 81± 22 keV b
S(300 keV) = 162± 39 keV b.
The relative error of the total S-factor of the results of [Oue+96; Kun+01; Fey04;
Ham+05] is in the range of 24% to 33%.
To improve these results, Schürmann et al. (2012) used a similar technique as
Ouellet et al. to analyze a large set of historical data. They used a FLRM for the
E1 and E2 ground state transition and a TLRM for the E3 ground state and the
cascade transitions in agreement with the elastic scattering data and the β-delayed
α spectrum of 16N. In addition, they calculated the systemic uncertainties for the
different data sets with the inclusion of a Monte Carlo simulation. They pointed
out that a major problem are the significant discrepancies in the various absolute
normalizations of the direct experiments. Their analysis results in [Sch+12]
S +8tot(300 keV) = 161± 19(stat)−2(sys) keV b.
Also, deBoer et al. (2017) analyzed a large set of historical data in consistency
with the elastic scattering data and the β-delayed α spectrum of 16N. For the fit,
they used a global R-matrix fit using the AZURE2 software [Azu+10; Ud15]. A
major innovation in their analysis is that they also calculated the uncertainties
arising from different models describing the S-factor. Their results are [deB+17]
S +18tot(300 keV) = 140± 21(MC)−11(model) keV b.
31
Chapter 2 - The Reaction 12C(α, γ)16O in the Helium Burning Stage in Stars
The relative error of the total S-factor of [Sch+12; deB+17] results in a range from
17% to 28%. To understand stellar evolution the S-factor in the low energy region
must be determined with a relative error of 10% or better. Although analysis and
measurements have improved in recent years the present relative error is far from
this desired uncertainty. The importance of the S-factor in predicting higher mass
elements will be briefly discussed in the next section.
Table 2.1: S-factor results at Ecms = 300 keV of the previous works of [DB74; Red+87;
Kre+88; Oue+96; Rot+99; Gia+01; Kun+01; Ham+05; Sch+12; deB+17]. The used
models to fit and extrapolate the data are R-matrix method (RM), K-matrix method
(KM), hybrid R-matrix method (HRM), three-level R-matrix method (TLRM), five-level
R-matrix method (FLRM) and the Monte Carlo (MC) simulation. More details can be
found in the text of Section 2.4. For a detailed description of all previous data please
refer to [deB+17].
S(300 keV)[keV b]
Reference E1 E2 Cascades Total Model
Dyer and Barnes (1974) 140+140−120 TLRM
80+50−40 HRM
Redder et al. (1987) 200+270 +24 a +0.5b−110 96−30 7± 2 , 1.3−1 TLRM; FLRM
140+120−80 80± 25 6± 3a HRM
Kremer et al. (1988) 0− 140 TLRM
Ouellet et al. (1996) 73.3± 13.2 TLRM
84.8± 15.7 KM
79± 16 36± 6 120± 40 KM; TLRM
2.4± 1d KM; TLRM
Roters et al. (1999) 79± 21 FLRM
Gialanella et al. (2001) 82± 16 RM; MC
2.4± 1d RM; MC
Kunz et al. (2001) 76± 20 85± 30 4± 4c 165± 50 TLRM; FLRM
Hammer et al. (2005) 77± 17 81± 22 162± 39 TLRM; FLRM
Schürmann et al. (2012) 83.4 73.4 4.4c 161± 19 +8(stat)−2(sys) FLRM; MC
7.9d FLRM; MC
deBoer et al. (2017) 86.3 45.3 7c 140± 21 +18(MC)−11(model) RM, MC
a 6.92MeV transition
b 7.12MeV transition
c all cascade transitions
d destructive inferference
32
2.4 Results of Existing Measurements
103
1974 Dyer
1987 Redder
1988 Kremer
1996 Ouellet
1999 Roters
2 2001 Gialanella10 2001 Kunz
2004 Fey
2006 Assuncao
2009 Makii
2012 Plag
AZURE2 Fit
101
100
10−1
0 0.5 1 1.5 2 2.5 3 3.5
Ecms[MeV]
(a) E1 contribution data [DB74; Red+87; Kre+88; Oue+96; Rot+99; Gia+01; Kun+01;
Fey04; Ass+06; Mak+09; Pla+12].
103
102
101
100 1987 Redder
1996 Ouellet
1999 Roters
2001 Kunz
10−1
2004 Fey
2006 Assuncao
2009 Makii
2011 Schürmann
2012 Plag
−2 AZURE2 Fit10
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Ecms[MeV]
(b) E2 contribution data [Red+87; Oue+96; Rot+99; Kun+01; Fey04; Ass+06;
Mak+09; Sch+11; Pla+12].
Figure 2.14: E1 (a) and E2 (b) contribution data fitted with the AZURE2
software [Azu+10; Ud15] configured with parameters of the supplemental ma-
terial provided by Ref. [deB+17]. The gray bar represents the Gamow window
together with the Gamow energy EG, which is shown as a blue dotted line.
33
SE2[keV b] SE1[keV b]
Chapter 2 - The Reaction 12C(α, γ)16O in the Helium Burning Stage in Stars
2.5 Production Factor
In this section, a brief summary of the production factor is given. The (isotopic)
production factor is a quantity to describe the ratio of isotopes produced in pre-
supernova evolution of massive stars with regards to nucelosynthesis. It is defined
as [Koz14; WHW02]
:= misoPiso ,
Xiso ·Mtot
with the mass fraction in the ejecta of a given isotope miso in solar masses, the
mass fraction of the isotope according to the solar metallicity Xiso, and the initial
mass of the star Mtot.
The key uncertainty in all models describing the pre-supernova evolution is the
S-factor of the reaction 12C(α, γ)16O [WH07; WW93]. Figure 2.15 shows the
calculated production factor as a function of the S-factor value. These results
were published in [WH07]. It is immediately visible that the production factor
strongly depends on the value of the S-factor. In addition the results of Schürmann
as well as deBoer are shown as gray bars in Figs. 2.15a and 2.15b. It is obvious
that no precise prediction for the production factor can be made based on the
present results. As discussed in the previous section, a relative uncertainty of
better than 10% for the S-factor of the reaction 12C(α, γ)16O is required to obtain
reasonable results for the pre-supernova evolution and the prediction of the mass
fraction of stars after the helium burning stage. For more information on the
production factor please refer to [WH07; WHW02; WW93; Koz14].
To improve the present results of the S-factor new measurements must be per-
formed. These measurements must fulfill two important criteria. First, the errors
in the range above 1MeV of the S-factor for the E1 and E2 ground state transition
must be substantially reduced. Furthermore, it is important to explore the range
below 0.9MeV. On the one hand, to rigorously exclude destructive interference
and, on the other hand, to reduce the range over which extrapolation is neces-
sary. Due to the strong decrease of the cross section from ∼ 10−11 b at 1MeV
to ∼ 10−17 b at 0.3MeV, the experiment must not only have a high luminosity
but also must be able to extract the background very accurately to determine the
S-factor in the region below 0.9MeV.
One of these upcoming experiments is MAGIX where processes with very low cross
sections at small momentum transfer can be studied. One goal of the MAGIX
physics program is the measurement of the electro-disintegration of oxygen to
determine the S-factor of the reaction 12C(α, γ)16O.
In the following chapter, we will have a closer look at the theory of electro-
disintegration. Subsequently, in Chapter 4, we will learn more about MAGIX
and will discuss the different phases of the planned measurement. At the end of
this thesis, we will discuss the results of the simulation of the electro-disintegration
cross section.
34
2.5 Production Factor
40
10
16O
18O
20Ne
23Na
24Mg
27Al
32S
36Ar
40Ca
Schürmann et al.
2
0 50 100 150 200 250 300 350
S(300 keV) [keVb]
(a) Isotopic production factor with the results of Schürmann et al. [Sch+12]
40
10
16O
18O
20Ne
23Na
24Mg
27Al
32S
36Ar
40Ca
deBoer et al.
2
0 50 100 150 200 250 300 350
S(300 keV) [keVb]
(b) Isotopic production factor with the results of deBoer et al. [deB+17].
Figure 2.15: The production factor of some key isotopes as function of the S-
factor. The data is from [WH07]. Two different results of S-factor calculations
are given in (a) and (b) to clarify the current situation with regards to the
uncertainties. The dotted blue line is the calculated S-factor and the gray bar
represents the error. Both results show that the current errors in the S-factor
calculations do not allow any robust prediction of the abundance of higher
elements that arise in star nucleosynthesis after the helium burning stage.
35
PPrroodduuccttiioonn FFaaccttoorr PPrroodduuccttiioonn FFaaccttoorr
36
CHAPTER 3
Electro-Disintegration Cross
Section of 16O
In this chapter, we will take a closer look at the theory of the electro-disintegration
reaction 16O(e, e′α)12C. The emphasis is on the derivation of the cross section.
The chapter concludes by showing how the cross section of the direct reaction
12C(α, γ)16O enters into the cross section of electro-disintegration.
In the electro-disintegration technique, electrons are scattered inelastically on oxy-
gen. By detecting the scattered electrons and the recoiled α-particles, the kinetic
parameters of the reaction can be determined. In contrast to photo-disintegration,
in which a γ-ray beam is used, a virtual photon is exchanged. In addition to
the transverse component, the longitudinal components and possible interferences
must also be considered in the electro-disintegration process.
We are using the natural system of units with ~ = c = 1. The following derivation
of the cross section is based on the procedure published in [FDM19; Don+17].
Nevertheless we have to keep in mind that the Extreme Relativistic Limit (ERL)
is not applicable for small electron scattering angles. Therefore, it is crucial that
formulas are used in its most general form, so that no approximations are done in
our calculations.
3.1 Kinematics of the Electro-Disintegration Reaction
In the electro-disintegration reaction of 16O (see figure 3.1), we define the four-
momentum of the virtual photon γ∗ as
qµ := (ω, ~q ) = Kµ −K′µ = Pµα + Pµ µ12 − PC 16 , (3.1)O
where ω is the energy and ~q is the three-momentum transfer:
ω = E ′e − Ee = Eα + E12 − EC 16 , (3.2)O
~q = ~k − ~k ′ = p~α + p~12 − p~C 16 . (3.3)O
37
Chapter 3 - Electro-Disintegration Cross Section of 16O
Kµ(E ,~k) K′µ(E′ ,~k ′e e )
e− e−
Pµ
∗ 12
(E12 , p~ )C C 12C
γ
12C
16O α
Pµ16 (E16 , p~16 ) P
µ
α (Eα, p~O O O α)
Figure 3.1: First order Feynman diagram for the electro-disintegration reaction
16O(e, e′α)12C.
By using the metric tensor gµν = diag(1,−1,−1,−1) we obtain
q qµµ = (Kµ −K′µ)2 = (Kµ)2 + (K′µ)2 − 2 · g νµνK K′µ (3.4)
= 2m2e − 2E E ′ + 2~∣k ·∣∣~ ′e e k ∣
= 2m2e − 2 ′ + 2
∣∣~ ∣EeEe k∣∣∣~ ′∣k ∣ cos(θe)
= 2m2e − 2EeE ′e (1− βeβ′e cos(θe))
=: −Q2,
where ∣∣∣ ∣~ ∣k∣
βe = =
|~ve|
,
Ee c
and θe the scattering angle of the electron. For the abso∣ lu∣ te values of the three-
momenta we will simply write q ≡ |~q |, px ≡ | ∣p~x| or k ≡ ∣~ ∣k∣.
3.2 Differential Cross Section of the Electro-Disintegration
Reaction 16O(e, e′α)12C
Following the Appendix B in [BD64] and (the equation (B1) for the differentialcross section of spinless particles and photons, the)cross section of the reaction
16O(e, e′α)12C in lab frame, where Pµlab = M ,~16 16 0 , can be written as:O O
1 ( )( )me M16 ∑ 3~ ′ 3 M d3p~dσ = O |M |2 · me d kfi ′ (2 )3 · Mα d p~α · 12C 12Cβ E 3 3e e E16 fi E(e π Eα (2π) EO 12 (2π)C ) (3.5)
·(2π)4δ4 Kµ + Pµ16 −K
′µ − Pµ µ
O α
− P12 ,C
whereM is the Lorentz invariant amplitude and∑fi represents the sum over final
and the average over initial spin states.
In an electro-disintegration experiment, the four-momentum of the scattered elec-
tron and the recoiled α particle are detected, but not the residual 12C. This
means we have to integrate the cross section over the momentum d3p~12 . WithC
the conservation of∫the three-m( omentum )
d3p~ δ3 ~k + p~ − ~12 16 k ′ − p~α − p~ = 1,C O 12C
38
3.2 Differential Cross Section of the Electro-Disintegration Reaction 16O(e, e′α)12C
we get
m2M M 1 1 ∑ k ′2d = e α 12C 2 dk ′dΩ′ep2αdpαdΩασ
E β (2π)5 |M | ·e e f( fi E ′i eEαE12C ) (3.6)
·δ E + E − E ′e 16O e − Eα − E12 .C
Furtheremore, we integrate the energy-conserving δ function over the recoil mo-
menta dpα. The integration will be carried out following the procedure described
in [GS64].
If we asume a function f(x) has any number of simple roots {x1, . . . , xn}, we can
write ∑n
( ( )) = δ(x− xi)δ f x
|f ′i=1 (
.
xi)|
In our case the function f(p)√in the lab fram√e is:
f(p) : = ω +M16 −√|p~ |2 +Mα −√ |~q − p~ |2 +MO 12C
= ω +M − p2 +M 2 216O α − q + p − 2p q cos θα +M12 ,C
where θα is the angle between ~q and p~α. According to the definition this function
has one simple root at p ≡ pα: √ √
f(p 2α) = ω +M16 − pα +Mα − p212 +MO C 12C
= E ′e − Ee + E16 − Eα − E12 .O C
The derivative of this functio∣n evaluated at p = p is∣ ∣ α ∣
′ ∣( ) ∣∣ = ∣∣∣∣√ −pf p + √−p+ q cos
∣
θα ∣∣∣
p=pα ∣ p2 +M∣ α |~q − p~ |∣
2 +M ∣12C
∣ p=pα= ∣∣−pα + −pα + q cos θαE ∣ E ∣
∣∣∣
α 12C
= 1 ∣∣ ∣∣Eαq cos θα − pα(Eα + E12 )∣E CαE12C
1 ∣∣∣ ∣∣= Eαq cos θα − pα(ω +M
E E 16
)∣.
O
α 12C
The integration o∫ver dpα results inp2αdp ( )α · δ Ee + E ′
E ∫E 16
− E
O e
− Eα − E12C
α 12C
= ∣∣∣ p
2dp
| | cos − ( + )∣∣ · δ(p− pα)Eα ~q θα pα ω M16 ∣O
= ∣∣∣ p
2
α ∣
Eα|~q | c∣∣os θα − pα(ω +
∣
M16 )∣O ∣
= pα · ∣∣∣1 + ωpα − Eα|~q | cos ∣∣∣
−1
θα
M ∣16 pαMO 16O
= pα · f−1
M rec
,
16O
39
Chapter 3 - Electro-Disintegration Cross Section of 16O
with the hadronic recoil factor frec. Finally, the differential cross section can be
expressed as
m2M M 1 k ′2 ∑d = e α 12σ C(2 )5 ′ · p f−1α rec |M |2fi dk ′dΩ′edΩα. (3.7)π M16 βe EO eEe fi
3.3 Calculation of the Matrix Element
Next, we will evaluate the Lorentz invariant matrix element ∑ 2fi|Mfi| , as de-
scribed in [Don+17, Section 7.3]. In the natural system of units e2 = 4πα, where
α is the fine structure constant, the invariant matrix element can be expressed as
[HM84; BD64]: ∑ ( )2
|Mfi|2 =
4πα µν
2 LµνWfi , (3.8)
fi Q
where Lµν describes the leptonic and W µν the hadronic tensor. With the electron
current jµ and the hadronic current matrix element Jµfi the leptonic and hadronic
tensors yield [Don+17, eq. 7.10-11] ∑
L = ∗µν ∑leptons jµjν , (3.9)
W µν µ∗ νfi = hadronsJfi Jfi. (3.10)
Both tensors, the lepton and the hadronic, are conserving the electromagnetic
current [HM84]
qµL νµν = q Lµν = 0, (3.11)
q W µνµ = qνW µν = 0. (3.12)
For unpolarized electron scat(tering the lepton (tensor corresp)ond)s to [HM84]:
Lunpolµν =
1
2 2 K
′
µKν +K′K ′ α 2ν µ − KαK −me gµν . (3.13)me
If we define
1 ( )
R ′µ = 2 Kµ +Kµ , (3.14)
we get
1( )
qµRµ = 2 Kµ +K
′
µ (Kµ −K′µ)
= 12(K
2 −K′2) = 12(m
2 −m2e e)
= 0.
We can then re-write the lepton tensor in terms of qµ and Rµ. Remembering
Eq. (3.4)
2K′ ααK − 2m2e = −(K2 − 2K′ ααK +K′2) = Q2,
and considering
4RµRν = (K ′µ +Kµ)(K +K′ν ν) = K ′µKν +KµKν +K K′ +K′ K′µ ν µ ν ,
q ′ ′ ′ ′ ′ ′µqν = (Kµ −Kµ)(Kν −Kν) = KµKν −KµKν −KµKν +KµKν ,
we get
2 1R ′ ′µRν − 2qµqν = KµKν +KµKν .
40
3.4 Definition of Spherical Unit Vectors
Finally, we get
2m2 1eLµν = 2RµRν − 2(qµqν +Q
2gµν). (3.15)
In the next step we will focus on calculating:
Rfi := 4m2e · L
µν
µνWfi . (3.16)
Using the current conservation[HM84]
q µµJfi(~q ) = 0,
we get ( )
Rfi = 2Rµ · 2R 2 µ∗ νν − qµqν −Q gµν · Jfi Jfi
= 2R Jµ∗ · 2R Jν − q Jµ∗q Jν −Q2µ fi ν fi µ fi ν fi J
µ∗ ν
fi gµνJfi
= |2R · J |2 −Q2fi |Jfi|2. (3.17)
To calculate (3.17), Jµfi(~q ) must be examined more closely. For a detailed approach
see the following section.
3.4 Definition of Spherical Unit Vectors
The hadronic matrix elements are three-dimensional Fourier transforms of the gen-
eral transition matrix elements of the ∫electromagnetic current operator [Don+17]:
Jµ (~q ) = dx eiqxJµfi fi(x).
The component µ = 0 is the Fourier transformation of the transition charge
density. The spatial components µ = 1, 2, 3 are the Fourier transformations of
the transition three-vector current den(sity [Don+17)]. Therefore, we can state
Jµ (~q ) := %(~q ), J~fi fi(~q ) .
For a set of unit vectors of the three dimensional space ex, ey, ez the eigenvectors
for the operators S2 and Sz are [Edm57]
e±1 = ∓√
1 (ex ± ie2 y
), (3.18)
e0 = ez. (3.19)
We obtain
S2em = 2em, for m = 0,±1
Szem = mem.
The set of em with m = 0,±1 is called spherical unit vectors and it defines an
angular momentum system with spin 1 [Edm57]. In addition the vectors have the
following properties [Edm57]:
1. The complex conjugate can be written as
e∗m = (−1)me−m for m = 0,±1.
41
Chapter 3 - Electro-Disintegration Cross Section of 16O
2. The vectors are orthogonal
e∗men = δmn for all m,n = 0,±1.
3. A vector V can be expressed in sph∑erical unit vectors as
V = Vme∗m.
m
where the components are given as scalar product
Vm = em · V.
The coordinate system we choose in this case is having its z-axis along the virtual
photon direction ~q and its y-axis normal to the scattering plane [Don+17]. In
more detail we choose
~q
ez(~q ) = , (3.20)|~q |
∣∣~k × ~k ′ey(~q ) = ∣ ∣ , (3.21)~ ∣k × ~k ′∣
ex(~q ) = ey × ez. (3.22)
In this specific coordinate system w∑e get
J~fi(~q ) = Jfi(~q ;m)e∗m(~q ), (3.23)
m
with
Jfi(~q ;m) = em(~q ) · J~fi(~q ).
Furthermore the current conservation implies [Don+17]
0 = q · Jµµ fi(~q ) = ω%fi(~q )− ~q · J~fi(~q )
= ω%fi(~q )− |~q |Jfi(~q ; 0).
Therefore, we can see a relationship bet(weeelements )n the longitudinal and charge matrix
ω
Jfi(~q ; 0) = %fi(~q ).
q
The representation of the electromagnetic current operator in spherical unit vec-
tors allows to further calculate the right-hand side of Eq. (3.17).
3.5 Calculation of Generalized Rosenbluth Factors
In this section we will determine Rfi by expressing it in terms of various combi-
nations of bilinear products of the current matrix elements:
RLfi := |% (~q )|
2
fi ,
RTfi := |Jfi(~q ; +1)|
2 + |Jfi(~q ;−1)|2,
RTLfi := −2 Re{%∗fi(~q ) (Jfi(~q ; +1)− Jfi(~q ;−1))},
RTTfi := 2 Re{J∗fi(~q ; +1)Jfi(~q ;−1)},
42
3.5 Calculation of Generalized Rosenbluth Factors
where the labels L, T, TL, and TT refer to the longitudinal, transverse, transverse-
longitudinal interference and transverse-transverse interference responses. They
are called nuclear response functions and contain all information of the nuclear
structure [Don+17].
Firstly we will calculate the two terms of Eq. (3.17) separately
2R Jµ = 2R J0 ~µ fi 0 fi − 2R · J~fi
= (E ′ ~ ~e + Ee)%(q)− 2R · Jfi. (3.24)
For the scalar product we use (3.23) to get
2R~ · J~ = (~k + ~k ′fi ) · J∑~fi(q)
= (~k + ~k ′) · Jfi(~q ;m)e∗m(~q )
m
= J (~q ; 0)(~k + ~k ′) · e∗fi 0(~q ) + Jfi(~q ; +1)(~k + ~k ′) · e∗+(~q )
+ Jfi(~q ;−1)(~k + ~k ′) · e∗−(~q ).
In a next step each of those scalar products will be evaluated separately
~ ~ ′ ~ ~ ′
(~k + ~k ′) · e∗(~q ) = (~k + ~k ′0 ) · ez(~q ) = (~ + ~ ′) ·
~q = (k + k ) · (k − k )k k
q q
k2 − k ′2 (E2 −m2= = e e)− (E
′ 2
e −m2e) E2= e − E
′ 2
e
q q q
= ω (E ′e + Ee).q
The coordinate system described in sec{tion 3}.4 implies that the y-axis is perpen-
dicular to the scattering plane = span ~k,~k ′ . This means (~k + ~k ′) · ey(~q ) = 0,
thus we get
(~k + ~k ′) · e∗ 1±1(~q ) = ∓√ (~k + ~k ′) · (ex(~q )− iey(~q )) = ∓√
1 (~k + ~k ′) · ex(~q )2 2
= ∓ 1√
2 · ∣∣∣~ × ~ ′∣∣∣(~k + ~k ′) · (~k× ~k ′)× ~q.q k k
We use the relation (a× b)× c = (a · c) b[− (b · c) a to evaluate ]
(~k + ~k ′) · (~k× ~k ′)× ~q = (~k + ~k ′) · k(2~k ′ +)k ′2~k − (~k · ~k ′)(~k + ~k ′)2
= 2k2k ′2 − 2 ~k · ~k ′ = 2k2k ′2(1− cos2(θe))
= 2k2k ′2 sin2(θe).∣ ∣
Additionally, considering the fact ∣∣~k× ~k ′∣∣ = kk ′ sin(θe) we finally get
(~k + ~k ′) · e∗±1(~q ) = ∓√
1 · 2k2k ′2 sin2(θ )
√2 · qkk
′ sin( ) eθe
= ∓ 2kk ′ sin(θe).
q
43
Chapter 3 - Electro-Disintegration Cross Section of 16O
When combining all theseparts we get( ) 2
2 ωR Jµµ fi(~q ) = (Ee + E ′e)1− %fi(~q )q
√
+ 2
[ ]
kk ′ sin(θe) Jfi(~q ; +1)− Jfi(~q ;−1)
q √ [ ]
= ( + ′ )Q
2 2
Ee Ee 2 %
′
fi(~q ) + kk sin(θe) Jfi(~q ; +1)− Jfi(~q ;−1) .
q q
To simplify the term we define
2
a := (Ee + ′ )
Q
Ee 2 , (3.25)
√ q
b := 2kk ′ sin(θe), (3.26)
q
and we get [ ]
2R µµJfi(~q ) = a · %fi(~q ) + b · Jfi(~q ; +1)− Jfi(~q ;−1) . (3.27)
This helps us to specify the first part of Eq. (3.17)
∣∣∣ ∣ [ ]22RµJµfi( ∣~q )∣ = a2|%fi(~q )|2 + ab%∗fi(~q )[Jfi(~q ; +1)− Jfi(~q ;−1)]
+ ab[% ∗ ∗fi(~q ) Jfi(~q ; +1)− Jfi(~q][;−1) ]
+ b2 J∗fi({~q ; +1)− J∗fi(~q ;−1) Jfi(~q ; +1)}− Jfi(~q ;−1)
= a2|%fi(~q )|2 + ab(Re %∗fi(~q )[Jfi(~q ; +1)− Jfi(~q);−1)]
+ b2 |J 2fi{(~q ; +1)| + |Jfi(~q ;−1})|2
− b2 · Re J∗fi(~q ; +1)Jfi(~q ;−1)
= a2RL − abRTL + b2RTfi fi fi − b2RTTfi . (3.28)
The second part of Eq. (3.17) becomes
∣∣∣ ∣2 ∑Jµfi(~q )∣∣ = J0( (~q )
2 − J~∗(~q ) · J~(~q ) = |%fi(~q )|2 − |Jfi(~q ;m)|2
m
ω2
) ( )
= 1− |% (~q )|2 − |J (~q ; +1)|22 fi fi + |Jfi(~q ;−1)|
2
q
= Q
2
RL2 fi −R
T
fi. (3.29)q
In summary we ge(t )
R = 2 − 2Q
2 ( )
a Q RLfi 2 fi + b
2 +Q2 RTfi − abRTL 2 TTq fi − b Rfi
=: v0vLRL Tfi + v0vTRfi + v0v RTLTL fi + v0vTTRTTfi , (3.30)
44
3.6 Calculating the Generalized Mott Cross Section
with the generalized Rosenbluth factors
( )(Q2)2
v v = (E + E ′ )20 L ( e e −)q
2
q2
, (3.31)
′ 2
v0vT = 2
kk sin(θ 2e) (+Q) , (3.32)q√ 2 ′
v v = − 2(E + E ′ Q kk0 TL ( e e) ) 2 sin(θe), (3.33)q q2
kk ′
v0vTT = −2 sin(θe) . (3.34)
q
By convention the common factor
v0 = (Ee + ′ )2Ee − |~q |
2, (3.35)
is separated. We will discuss its importance in the following section. We also
define ρ := Q22 = 1− ω
2
2 which implies 0 ≤ ρ ≤ 1 [Don+17, Section 7.1].q q
3.6 Calculating the Generalized Mott Cross Section
Before combining all results to describe the differential cross section it is necessary
to determine the Mott cross section in its general form. As stated in [BD64,
Section 7] the differential Coulom(b scattering cross sectio)n can be written asdσ 2
dΩ′ =
α
4 4E
′ µ ′ 2
eEe − 2K Kµ + 2me .
e Q
When using the relativistic energy-momentum relation m2e = E2 − k2e we get:
4EeE ′e − 2KµK′µ + 2m2e = 2E E ′e e + 2~k · ~k ′ + 2m2e
= E2e − k2 + E ′ 2 ′2e (− k +) 2E ′eEe + 2~k · ~k ′2
= (E ′e + Ee)
2 − ~k − ~k ′ = (Ee + E ′e)
2 − (~q )2
= v0.
And so the generalized(Mott)cross section is
dσ = dσ · E
′
e = α
2v E ′0 e
dΩ′ dΩ′ 4 . (3.36)e Mott e Ee Q Ee
3.7 Combining Results to Differential Cross Section
As consequence of the previous d(erivat)ion, we can write the invariant matrixelement now as ∑ 2
|M |2 = 4πα
∑
fi 2 ·
v0
Q 4m2 vKRKfi e K
with K = L, T, TL, TT and then the c(ross s)ection be(comes )
MαM m
2 2
d 12 e 1 k
′2 4πα v0 ∑
σ = C −1(2 )5 · ′ pαfrec (2 · 4 2) v R dk
′dΩ′K K edΩαπ M16 βe E EO e e Q me K
M M β′2 ′ 2= α 12
∑
C · e Ee α v0 −1 ′ ′8π3 · 4 · pαfM β E Q rec vKRK dk dΩedΩα.16O e e K
45
Chapter 3 - Electro-Disintegration Cross Section of 16O
Finally, we get the differential cross section for the electro-disintegration calculated
in the lab frame
( )
dσ M M ′2= α 12C · βe · dσ −1
∑
dk ′dΩ′ dΩ 8 3 dΩ′ · pαfrec · vKRK . (3.37)e α π M16 βO e e Mott K
3.8 Transformation to Center of Mass Frame
Instead of calculating the differential cross section (3.37) in the lab frame of the
experiment dΩlabα it is useful to calculate the hadronic part in the center of mass
system ∑(cms) of the outgoing carbon-α pair. The goal is to express the term
p f−1α rec · K vKRK in the cms. The transition from the lab to the cms includes a
Lorentz boost along the virtual photon three-momentum ~q.
The invariant mass W of the o∣utgoing ca∣ rbon∣ -α system∣ is defined as
2 ∣∣ µ µ ∣∣2 ∣∣ µ µ∣∣2W := q + P16 = PO 12 + PC α .
In the lab frame the following applies ( )
Pµ16 +
µ = ω +Mq 16O ,
O ~q
and so W 2 = (ω 2 2lab +M16 ) − qO lab.
The right side of
plab ∫ 2 ( )α · f−1 pαdpα ′
M rec
= · δ Ee + E16 − EO e − Eα − EE E 12 ,C16O α 12C
needs to be evaluated in a specific fra(me )and we will choose the cms:
Pµ + Wqµ = = Pµ + Pµ16 .O ~0 12C α
The energy and momentum conservation yields W = E16 + ω = E12 + Eα, and
p = p =: pcms
O C
α 12 α . So, definingC √ √
f(p) := W − p2 +M2 2 2α − p +M12 ,C
we have
∂f ∣∣ ∣∣∣ ∣∣∣∣pcms
∣
= α + p
cms ∣
α ∣∣∣ cms= Wpα .∂p p=pcms Eα E12 EαEα C 12C
The integration results in
plab ∫ p2 dp ( )α · αf−1 αrec = · δ Ee + E ′M ∫ E E 16
− E
O e
− Eα − E12C
16O α 12C
= p
2 dp cmscms pα
Wpcms
δ(p− pα ) = .
α W
This means in particular
plab
M16
f−1 = O pcmsα rec W α . (3.38)
46
3.8 Transformation to Center of Mass Frame
To calculate vK and v0 in the cms we have to boost these scalar parameters along
~q.
For a relative velocity ~vrel = |~vrel|·n̂ the general Lorentz boost of a four-momentum
P µ = (E, p~ ) can be written as [DL05]
Eboost = γrel(E − βrel(p~ · n̂)),
p~ boost = p~+ (γrel − 1)(p~ · n̂) · n̂− γrelβrelE · n̂,
with βrel = vrel and −1γ 2rel = [1− βrel] . If p~ = p~‖+p~⊥ with p~⊥ ·~v = 0 and p~·~v = p~‖ ·~vc
we have
p~boost‖ = γrel(p~ · n̂− βrelE) · n̂,
p~boost⊥ = p~⊥.
The Lorentz boost into the center of mass system means a boost along ~q. The
velocity vector is given as ~vrel = 1 ~q, so we have:Wγrel
q
βrel = + ,ω M16O
ω +M16
γ Orel = ,
W
γrel·βrel =
q
.
W
Boosting the three-momen(tum of the vir)tual photon we get~q lab ( lab)
~q cms = γ qlab − β ωlab = − ωγ ~q labrel rel lab relq W
M
= 16O ~q lab.
W
To boost the scalar parameter, let’s also have a closer look at the energy of the
incoming and outgoing( electron )
~ lab lab
Ecms lab
βrel ~ lab lab lab k · ~q
e = γrel Ee −( )· k · ~q = γrelEq e − W
k lab
2
− ~klab · ~k ′lab
= γ Elabrel e − ,W ( )
~ lab ~ ′lab ′lab 2
E ′ cms = γ E ′ lab
k · k − k
e rel e − .W
Summing it up, we(get ) lab 2 ′lab 2 ( lab)( )
Ecms + E ′ cms = γ Elab + (k ) − (k ) ωE ′ lab − = γ − Elab + E ′ labe e rel e e relW W e e
M ( )
= 16O Elabe + E ′ labe .W
This enables us to calculate ( )2( ) ( )2( )
vcms = (Ecms ′ cms 2 cms 2
M16 lab ′ lab 2 M 2
0 ( e +)Ee ) − ( ) =
O + − 16q E E Oe e qlabW W
M 2
= 16O vlab
W 0
.
47
Chapter 3 - Electro-Disintegration Cross Section of 16O
Additionally the substitution a Eq. (3.25) in the cms can be expressed as:
2
acms = (Ecms + E ′ cms Qe e )(qcms)2
= W alab.
M16O
The substitution b Eq.((3.26))can be written as(: )
b := ~k + ~k ′ · 1e∗−1(~q ) = −√ ~k + ~k ′ · ex(~q ).2
Since ex(~q ) ⊥ ~q we get
1 ( )
b = −√ ~k + ~k ′ · ex(~q ),2 ⊥
from which we directly obtain bcms = blab, because the boost of the orthogonal
part doesn’t change. Inserting this in the formula to calculate vcmsK we finally get
2
vcms
M
= 16O vlab0 2 0 ,W
4 2
vcms = W lab cms W labL M4 vL , vT = 2 vT ,16 MO 16O
3
cms = W lab cms W
2
v v labTL M3 TL
, vTT = M2 vTT .16O 16O
M2
Defining 2ṽ := W vcms = vlab and ṽ := 16O vcms, we have ṽ ṽ = vcmsvcms0 2 0 0 K 2 K 0 K 0 K .M16 WO
This enables us to write the differential cross section of the electro-disintegration
with the hadronic part calculated in the cms of the carbon-α system
dσ M M ′2α 12C βe cms ∑ cms
dωdΩ′ dΩcms = 8 3 · · pα σMott · ṽKRK , (3.39)e α π W βe K
with
α2vlab ′ lab
σ = 0 EeMott
Q4 Elab
,
e
and
= W
2
lab WṽL 2 vL , ṽTL = v
lab,
M16 M
TL
O 16O
ṽ = vlabT T , ṽTT = vlabTT .
3.9 Approximation Using the Extreme Relativistic Limit
It is also usefull to determine the approximated formulas based on the ERL. This
is applicable for high electron energies Ee  me with scattering angles θe ≥ 1°.
These can be used, for example, for simulations or analyses in this parameter
range, since the calculations with these approximated formulas are significantly
more time efficient.
By using the ERL, the four-momentum transfer Q2 Eq. (3.4) can be expressed as:
Q2 = 2E E ′e e(1− cos(θe)).
48
3.9 Approximation Using the Extreme Relativistic Limit
Using the double-angle formula 2 sin2(θ) = 1−( co)s(2θ) we get
Q2 = 4E E ′ sin2 θee e 2 . (3.40)
At this point we also want to calculate the generalized Rosenbluth factors using
the Extreme Relativistic Limit.
Using the ERL on Eq. (3.35) we get
v = (E + E ′ )2 − q2 = E2 − k2 + E ′ 2 − k ′20 e e e e + 2EeE ′e + 2kk ′ cos(θe)
= 2E E ′e e(1 +(cos()θe))
= 4 θeEeE ′e cos2 2 .
Per definition the longitudinal part c(an b2)e expressed as2
vL =
Q
2 = ρ
2.
q
We can use Eq. (3.40) to simplify ( ) ( )
2 ′2 sin2( ) = 4 2 ′2 sin2 θek k θe k k 2 cos
2 θe
2
1
2 ( =) v Q
2
4 0 ,
Q = tan2 θe
v0 2
.
Therefore, we get
2 ( )1 Q θe
v 20vT = 2v0 2 + v0 tan ,q 2
and finally ( )
1 θe
v 2T = 2ρ+ tan 2 .
For the transverse-longitudinal interference we have
√
2 ′
vTL = − · (
kk
Ee + E ′e)ρ · sin(θ(e)v0 q√ ) ( )
2 (E + E ′ρ e e)EeE ′e · 2 sin( θ2)e cos θe= − 2
q 4E(eE ′)cos2 θee 2
′
= −√ρ
(Ee + Ee() sin) θe2 .
2 q cos θe2
If we take a close(r lo)ok at( the denominator of)the r(ight)fraction we can express
(E + E ′ 2 2 θe ′ 2 ′ 2 θee e) sin 2 = (Ee −(Ee))+ 4EeEe sin (2 ) ( ( ))
= 2 sin2 θe + 2 = 2 sin2 θe + 2 1− sin2 θeω ( 2) Q q( ) 2 Q 2
= q2 sin2 θe +Q2 cos2 θe2 2 .
49
Chapter 3 - Electro-Disintegration Cross Section of 16O
Finally, we get √√√ ( )
vTL = −√
ρ √ θeρ+ tan2
2 2
.
Similar to the calculations we performed for the transverse part, the transverse-
transverse interference part results in
= −1vTT 2ρ.
In summary, we achieve
( )
v0 = 4E ′eEe cos2
θe
2 (3.41)
v 2L = ρ , ( ) (3.42)
1 θe
vT = 2ρ+
2
√tan 2 , (3.43)
ρ √√√ ( )vTL = −√ ρ+ tan2 θe2 , (3.44)2
= −1vTT 2ρ. (3.45)
All of these results are in agreement with those of [Don+17].
The Mott cross section can be simplified with the ERL
( ) 2 ′ 2 ( )dσ 4α Ee
dΩ′ = 4 cos
2 θe .
e Mott Q 2
That is equivalent to the Mott cross section, that one can find for example in
[Pov+15]. When using Eq. (3.40) we get
( ) ( ( ))2
dσ
dΩ′ = ( α cos
θe
2 )2 ,
e Mott 2E sin2( θee 2 )
which is equivalent to the Mott cross section, that one can find for example in
[Don+17].
3.10 Multipole Decomposition of the Response Functions
To determine the nuclear response functions it is necessary to make some assump-
tions. First of all, we assume that the inital oxygen nucleus is in the ground state
J = 0. Furthermore we assume that the recoil nuclei are also in the ground state.
In addition, we will restrict the consideration to the electric and Coulomb mul-
tipoles up to the quadrupole contribution. Following the development in [RD89;
FDM19] the response functions RK in the cms can be written in terms of Legendre
polynomials Pk(cos θα) and associated Legendre polynomials P nk (cos θα)
50
3.10 Multipole Decomposition of the Response Functions
( )
R = P (cos θ ) |t |2 + |t |2 + |t |2L 0 α  C0 C1 C2 √ (3.46)
√
+P1(cos θ α) 2 3|tC0||tC1| cos(δC1 − δC0) + 4 35 |tC1||tC2| cos(δC2 − δ( ) C1
)
+ 10
√
P2(cos θ ) 2|t |2α  √C1 +
2
7 |tC2| + 2 5|tC0||tC2| cos(δC2 − δC0)
+ (cos )6 3 ( )P3 θα 5 |tC1||tC2| cos( − )+ (cos ) 18δ 2( ) C2
δC1 P4 θα 7 |tC2| ,
RT = P 2 20(cos θα)(|tE1| + |tE2| ) (3.47)
+P1(cos
6
θα)(√ |tE1||tE2| cos(δ)E2 − δE1)5
+P2(cos θα)(−|tE1|
2 + 57 |t
2
E2| )
+ 6P3(cos θα)(−√ |tE1||512 )
tE2| cos(δE2 − δE1)
+P (cos θ ) − |t |24 α E2 ,
 ( 7
R = P 1 √TL 1 (cos θα) 2 3|tC0||tE1| cos(δE1 − δC0) (3.48)√
− 2 35 |tC2||tE1| cos(δC2 − δE1))
(+
√6 |t
5 C1
||tE2| cos(δC1 − δE2)√
+P 12 (cos
5
θα) 2|tC1||tE1| cos(δC1 − δE1) + 2 3)|tC0||tE2| cos(δE2 − δC0)
1√0
(+√ |tC2||tE2| cos(δ7 3 C2 − δE2) )
+P 13 (cos ) 2
3 4
θα 5 |tC2||tE1| cos(δ( C2
− δE1) +√ |tC1||tE2| cos(δ) 5 C1
− δE2)
√ 
+P 14 (cos )
6 3
θα 7 |tC2||tE2| cos(δC2 − δE2)  · cosφα,
RTT =−RT cos(2φα), (3.49)
with tC,EJ representing the reduced Coulomb and electric matrix elements and
δC,EJ the phases of each multi-pole operator. Both are functions of q and ω
[FDM19]. Using the Rodrigues’ Formula the Legendre Polynomials can be repre-
sented as [AS74]
1 dn( 2 − 1)n
Pn(
x
x) = 2 ! ( d ) , (3.50)nn xn m m
Pmn (x) = (−1)
m 1− 2 2 d Pn(x)x d . (3.51)xm
In Appendix A the nuclear response functions up to octupole contributions in
terms of the Legendre and the associated Legendre polynomials can be found.
The nuclear response functions are also given there in terms of sine and cosine.
51
Chapter 3 - Electro-Disintegration Cross Section of 16O
3.11 Representing the Electric Matrix Elements tEJ with
the S-Factor Components SEJ
The differential cross section for the photo-disintegration can be established with
similar calculat[ions as ]done for(the electr)o-disin(tegrat)ion cross section [FDM19]
dσ M cms= α
M12C pα α cms
dΩcms 4πW ~c Ecms RT (γ, α), (3.52)α (γ,α) γ
with Ecmsγ = ωcms = |~qcms| and th(e transve)rse nuclear response function RcmsT (γ, α).
In the lab frame, where Pµlab16 = √M ,~16 0 , the invariant massW can be expressedOas O
W = M (M + 2Elab16O 16O γ ).
For the limit Q2 → 0, the transverse nuclear response function in electro-disinte-
gration is the same as the one in photo-disintegration [FDM19]. This allows us to
establish a connection between the electric matrix elements tEJ and the S-factor
components SEJ . In the following the transverse nuclear response function in cms
is simplified as RT .
First, we use the reciprocity theorem Eq. (2.32) to represent the response function
in terms of the S-factor. In the center of mass frame of the carbon-α system we
have pcms = qcms = Ecms16 γ . As we have l = 0 for all initial and final particles theO
reciprocity equation for the photo-disintegration reaction is
σ(γ, α) = 1 |p~
cms 2
α |
σ(α, γ) 2 |Ecms .γ |2
So, the differential[cross se]ction of(the α-cap)ture reaction ca be written as
dσ M M Ecms
( )
= α 12C γ αdΩcms 2 cms RT . (3.53)γ ( πW ~c pα,γ) α
Finally, the transverse nuclear response function [in cms ]becomes
= ~c · p
cms
α ·W dσRT 2α · Ecms · 4π . (3.54)γ ·M cmsαM12C dΩγ (α,γ)
By comparing the coefficients of equations Eq. (2.28) and Eq. (3.54), the electric
matrix elements in the limit Q2 → 0 can be expressed as functions of the S-factor
components:
√
= √√
√
~c · p
cms W S (Ecms)e−2πηα · · E1t αE1 2 cms , (3.55)α Eγ MαM12 EcmsC α
√√√√ ~c pcms cms −2πηt = · α W SE2(Eα )eE2 2 · ·α Ecms M M Ecms , (3.56)γ α 12C α
where the Sommerfeld number depends on the cms energy η ≡ η(Ecmsα ).
52
CHAPTER 4
Accelerator and Experimental
Setup
In this chapter we will discuss the entire experimental setup for the planned
measurement of the S-factor of the reaction 12C(α, γ)16O. First, we will get a
brief overview of the electron accelerator – MESA. Then, we will get a broad
overview of the experiment MAGIX located at MESA. For this purpose, we will
take a look on all important components of MAGIX at MESA that are required
for the S-factor measurement. At the end of the chapter the different phases of
the planned measurements are summarized.
4.1 Mainz Energy-Recovering Superconducting Accelera-
tor
MAGIX
Beam Dump
Hydro-Møller
Injector
MAMBO
MAGIX
P2 STEAM
MARC 2, 4
Cryomodules
5MeV MELBA
Beam Dump
MARC 1, 3, 5
Figure 4.1: Overview of the structure of MESA [Sim21b; Bec+18].
53
Chapter 4 - Accelerator and Experimental Setup
The Mainz Energy-Recovering Superconducting Accelerator (MESA) (see Figs. 4.1
and 4.2) is an electron accelerator that is currently under construction at the Jo-
hannes Gutenberg University in Mainz, Germany. It will be a low-energy con-
tinuous wave recirculating electron linac for various particle and nuclear physics
experiments [Hug+20; Hug+17]. Figure 4.1 shows the structure of MESA. The
first part of MESA is the injector. The photo-source STEAM (Small Thermalized
Electron Source At Mainz) supplies a polarized or an unpolarized electron beam
with an energy of 100 keV and a beam current of up to 1mA. The photo-source is
followed by the low-energy beam transport system MELBA (MESA Low–energy
Beam Apparatus). MELBA contains a spin manipulation system and a chopper-
buncher section for longitudinal matching. This acts as a beam preparation for
the normal-conducting booster linac MAMBO (Milliampere Booster) [Hug+20].
MAMBO accelerates the electrons to an energy of 5MeV. Then, the electron
beam is directed into the main linac. The main linac consists of two accelera-
tor units, the cryomodules MEECs (MESA Enhanced ELBE Cryomodules). The
electrons are boosted up to 25MeV per cryomodule. Depending on the mode and
maximum beam energy, the electrons are transported through the recirculation
arcs 1-5, MARCs (MESA Arcs) 1-5. With this method, the electron beam can
be boosted several times in the two MEECs [Sim21b; Led20; Hug+20; Hug+17].
MESA can be operated in two different operating modes.
In the External Beam (EB) mode MESA provides a polarized beam with an
energy of up to 155MeV and a current of 150µA [Hug+20; Bec+18]. For this
purpose, the beam is accelerated up to 3 circulations using MARCs 1-5. With
each circulation the beam gains a total energy of up to 50MeV. Once the beam
has reached its final energy, it is extracted from the accelerator part and guided
through the Hydro-Møller to the P2 experiment. The Hydro-Møller measures the
polarization of the electron beam [Kem20]. After the P2 experiment, the beam is
dumped in a dedicated beam dump [Sim21b; Hug+20; Bec+18].
In the Energy Recovering (ER) mode a polarized or an unpolarized electron beam
with an energy of up to 105MeV and a current of 1mA is used for the internal
target experiment – Mainz Gas Injection Target Experiment (MAGIX). For this
purpose only MARC 1-3 are used and the beam will be extracted from the ac-
celerator part after a maximum of 2 recirculations. Downstream of MAGIX the
beam is guided back to the accelerator part with a phase shift of 180°. Due to
this phase shift, the beam transfers its energy back to the cryomodules and is
thereby decelerated. Up to 25MeV can be returned to the MEEC each pass with
this method. After a maximum of 4 recirculations, the remaining 5MeV electron
beam is stopped in the 5MeV beam dump. This energy recovery allows the high
electron current of 1 MeV. In the second expansion stage of MESA, 10 mA elec-
tron beam current should also be possible in ER mode [Sim21b; Led20; Hug+20].
Instead of using the energy recovery, the electron beam can also be dumped after
MAGIX. For energies up to 30MeV a current of up to 1mA can be achieved. At
higher energies, the beam dump can only be used if the beam current is reduced.
This is necessary on the one hand because otherwise the energy demand would be
too high without ER mode, and on the other hand the beam dump is not designed
for such a high radiation exposure [Sim21b, section 11.2 & 13.1].
For a full description of MESA, the components and the two modes please refer to
the PhD thesis [Sim21b]. For further information also refer to [Hug+20; Led20;
Hug+17].
54
4.1 Mainz Energy-Recovering Superconducting Accelerator
P2
MAGIX
DarkMESA
Figure 4.2: Overview of the MESA hall [Sim21a] and the experiments P2
[Sor17], DarkMESA [Chr21b] and MAGIX [Sch+21]. The dashed line intends
to clarify the alignment of P2 and DarkMESA. P2 and DarkMESA carry out
their measurements in EB-mode. In ER mode MAGIX will carry out its
measurements.
Cherenkov Detector
Solenoiding
Supercon
duct
Tracking Planes beam
dump
Liquid Hyrdrogen Target
−
ized e b
eam
polar
Vacuum Chamber
Shielding
Figure 4.3: Overview of the P2 experiment [Bec+18; Sor17; Ber16]
4.1.1 The P2-Experiment
The P2 experiment plans to measure the weak mixing angle θW , which is also
known as Weinberg angle. For this purpose, the parity-violating asymetry of the
elastic electron-proton scattering is determined. Figure 4.3 shows the layout of the
55
Chapter 4 - Accelerator and Experimental Setup
P2 experiment. The longitudinally polarized electron beam scatters on the liquid
hydrogen H2 target. The scattered electrons are focused towards the Cherenkov
detector by the P2 spectrometer magnet. To measure the scattering angle, the
track of the electrons can be reconstructed by detecting them in the tracking
planes. The shield located radially around the outgoing beam is used to remove
the background created by bremsstrahlung or Møller scattering. After the P2
experiment the beam will be dumped. [Ber21; Kem20; Bec+18; Sor17; Ber16].
4.1.2 The DarkMESA Experiment
Veto-System Pb glass Calorimeter
PbF2 Calorimeter
Figure 4.4: Overview of the DarkMESA experiment [Chr21b].
The parasitic dark sector experiment DarkMESA will use the high-power beam
dump of the P2 experiment as target. DarkMESA (see Fig. 4.4) was developed
to detect Light Dark Matter (LDM). In the simplest model, the LDM couples
to a massive vector particle, the so-called dark photon. This can potentially be
generated in the P2 beam dump by a process analogous to bremsstrahlung. The
dark photon then decays into a pair of dark matter particles. In the DarkMESA
calorimeter, a fraction of these dark matter particles can scatter on electrons
which can then be detected. The experiment is placed outside of the accelerator
hall well shielded from beam-related SM particles. To get rid of any kind of
remaining background, a versatile veto system is built around the calorimeter.
The beam time of the DarkMESA experiment is coupled to that of P2, which
means that DarkMESA can take data while P2 is operated in EB mode. In
addition, DarkMESA can carry out background measurements when MESA is
switched off [Chr21b; Chr21a; Chr+20a; Chr+20b; Dor+19; Dor+18].
4.2 Mainz Gas Injection Target Experiment
The Mainz Gas Injection Target Experiment (MAGIX) is a versatile fixed-target
experiment, which will be operated at the new electron accelerator MESA in Mainz
(see Figs. 4.2 and 4.5). One of the core components of MAGIX is the internal gas
jet target [Gri+18; Gri18; Aul21] which allows the production of a directional gas
56
4.2 Mainz Gas Injection Target Experiment
Jet Target
Scattering Quadrupole
PORT Chamber STAR
Dipoles
GEM-based
TPC
Trigger-Veto-
System
Shielding
Figure 4.5: Overview of the MAGIX experiment. The gas jet target will be
mounted in the center of the scattering chamber. A pumping station will
remove the injected gas. Two identical magnetic spectrometers STAR and
PORT will be able to rotate around the interaction point. The spectrometers
will be connected to the vacuum of the scattering chamber. They will focus
the scattered electrons with a quadrupole-dipole-dipole configuration onto the
focal plane located in the GEM-based TPC. At the entrance to the TPC, the
scattered electrons will pass for the first time through a vacuum termination
foil. After the TPC, a veto trigger system will be mounted for triggering,
timing and particle identification purposes [Sch+21].
beam through the vacuum of the scattering chamber with an areal particle density
of >1018 atoms/cm2. With the combination of the low-density target and the high
beam current, MAGIX will reach a luminosity of O(1035 cm−2 s−1). To determine
the electron scattering kinematics with a high momentum and angular resolution,
MAGIX will use two magnetic spectrometers equipped with GEM-based TPCs
in the focal planes (see Fig. 4.5). This will allow to study processes with very
low cross sections at a small momentum transfer. The rich physics program of
MAGIX ranges from low-energy nuclear physics up to new physics searches, like
[Sch+21]
• Measurement of electromagnetic form factors, charge radii, e. g. proton ra-
dius;
• Search for particles of the dark sector [Mer+11];
• Reaction studies of astrophysical relevance, e. g. 12C(α, γ)16O;
• Nuclear few-body physics experiments.
In the following we will get a brief overview of the different parts of MAGIX. We
will focus on the components necessary for measuring the S-factor of 12C(α, γ)16O.
57
Chapter 4 - Accelerator and Experimental Setup
4.2.1 Gas Jet Target
Laval Nozzle
Gas Jet
Beam
Catcher
(a) Sketch of the gas jet nozzle and
catcher principle (b) Image of the current nozzle and
catcher
Figure 4.6: The left sketch (a) shows the direction of the gas jet and the
interaction with the electron beam. The picture on the right (b) shows the real
nozzle and catcher used in the current test stand [Aul21].
The basic idea of the MAGIX target is to scatter electrons on gas that passes
through the vacuum chamber without breaking it. For this purpose a thin, di-
rected gas jet is generated by a Laval nozzle. Due to the hydrodynamics of the
Laval nozzle, the gas becomes a directed laminar flow in the scattering chamber.
This laminar flow is stable for a few millimeters. The gas jet is then sucked off
by a catcher placed closly behind the nozzle. The electron beam does not pass
any window which is an advantage of such a system. The background is reduced
significantly in comparison to other fixed target experiments where for instance
target walls are inducing multiple scattering. Figure 4.6 shows a schematic sketch
of the nozzle and catcher as well as the gas jet.
To ensure a constant areal particle density, the pressure and temperature in front
of the nozzle and the volume flow must be controlled. In addition, the design of
the Laval nozzle, especially the narrowest inner and the outlet diameter, play a
decisive role. The calculations in [Gri+18] show that the stagnation conditions
with the lowest temperature-pressure combination should be selected. This allows
MAGIX to be carried out with a low volume flow.
The MAGIX Gas-Jet Target (see Fig. 4.7) has been developed by the University
of Münster, Germany. It essentially consists of a cold head in which the gas
is adiabatically cooled. The cold head has two cooling stages, the warm stage
can cool down to 28K and the cold stage can cool down to 8K. The nozzle is
mounted directly behind the cold stage. In order to protect the cooling stages
against thermal influences, the cold head is surrounded by an insulation vacuum.
For a large volume flow, the pre-mounted booster stage can pre-cool the gas down
to 77K. An areal particle density of >1018 atoms/cm2 is planned for the MAGIX.
For hydrogen this can be achieved by cooling the gas with a gas flow of 2400L/h to
40K. Two different nozzles with an inner diameter of 0.5mm were also designed
by the University of Münster, Germany. For hydrogen, an areal particle density
of 5× 1018 atoms/cm2 is calculated for the nozzle with an outlet diameter of 1mm
58
4.2 Mainz Gas Injection Target Experiment
1430mm
Booster Stage
Cold Head
Insulation Vacuum
Warm Stage
Cold Stage
Nozzle
Figure 4.7: Overview of the MAGIX gas jet target. With the cold head in
combination with the booster stage hydrogen at a gas flow of 2400 l/h can be
cooled down to 40K to achieve an areal particle density of >1018 atoms/cm2.
In addition to hydrogen, helium, argon, oxygen, nitrogen, xenon and various
other types of gas can be used. The nozzle can easily be exchanged to adapt to
the respective gas. [Gri18; Gri+18; Aul21].
and an areal particle density of 2.4× 1018 atoms/cm2 with an outlet diameter of
2mm [Gri+18]. In addition to hydrogen, helium, argon, oxygen, nitrogen, xenon
and various other types of gas can be used. The nozzle can easily be exchanged
to adapt to the respective gas [Sch+21; Gri+18; Gri18; Aul21].
At the A1 experiment at Mainz Microtron (MAMI), the MAGIX Gas-Jet Target
has been commissioned in 2018 and 2019 to carry out first experiments. During
these measurements it was also possible to calculate the areal particle density.
With a hydrogen flow rate of 2400 l/h at a temperature of 40K with the nozzle of
1mm outlet diameter a maximum areal particle density of 1.6× 1018 atoms/cm2
was achieved [Sch+21]. Further information on these measurements can be found
in [Sch+21].
59
Chapter 4 - Accelerator and Experimental Setup
4.2.2 STAR/PORT Spectrometer Setup
Interaction
PORT Point STAR
Dipoles Quadrupole
Focal Plane
GEM based
TPC
Veto-Trigger-
System
Figure 4.8: Overview of the MAGIX spectrometer. The spectrometers are
connected to the vacuum of the scattering chamber. Each of the two identical
magnet-spectrometers focuses the scattered electrons with a quadrupole-dipole-
dipole configuration on the focal plane. The focal plane is located in entrance
of the GEM-based TPC. After the TPC, a veto trigger system is mounted for
triggering, timing and particle identification purposes [Sch+21].
To detect the scattered electrons or recoiled positrons, MAGIX uses two identical,
high-resolution magnetic spectrometers, called PORT and STAR (see Fig. 4.8).
The spectrometers are named after the nautical terms port and starboard and are
colored with their respective colors, red and green. Because of the limited height
of the hall and the relatively high position of the beamline, the spectrometers had
to be oriented upside down. They are mounted on opposite sides of the beam line
on a system that allows the spectrometers to be rotated around a common pivot
point. The adjustable angle range per spectrometer relative to the beam line is
15° to 165°. The spectrometers are connected to the scattering chamber via a
sliding seal so that they share the same vacuum. This means that the scattered
electrons and recoiled particles enter the spectrometer without passing through
any foil. By reversing the polarity of the magnets, the particle species (electron
or positron) can be selected.
The quadrupole-dipole-dipole configuration of the spectrometer maps the abso-
lute momentum, the scattering angles, the azimuthal angle and the vertex to the
Cartesian coordinates x and y of the focal plane, and the corresponding entrance
angles at the focal plane. Thus, it is possible to replace the direct measurement
of these quantities with the potentially simpler position measurement in the focal
plane. For this purpose a GEM-based TPC located in the focal plane is used. For
60
4.2 Mainz Gas Injection Target Experiment
example, the absolute momentum is proportional to the x-coordinate in the focal
plane to the first order.
At the entrance of the TPC is the first vacuum foil, which the particles must pass.
With this setup, MAGIX will be able to determine the scattering parameters with
high resolution. Due to the small material budget, effects such as multiple scat-
tering will be reduced to a minimum. From the three-dimensional reconstructed
track of the TPC in combination with the trigger, timing and particle information
of the trigger veto system, the scattering parameters such as momentum, polar
and azimuthal angles, and interaction point can be determined with high precis-
sion. The goal is to achieve a relative momentum resolution of ∆p < 10−4 and an
p
angular resolution of ∆Θ < 0.05° [Sch+21; Sch21].
Time Projection Chamber
In the focal plane of each spectrometer, the electrons or positrons can be detected
with a Time Projection Chamber (TPC). Figure 4.9 illustrates the working prin-
ciple of a TPC. When a charged particle passes through the active gas volume of
the TPC, the counting gas is ionized. The free electrons drift along the external
electric field towards the amplification stage, where several Gas Electron Multi-
pliers (GEMs) are located. The x- and z-coordinates of the electrons read out
at the pad plane anode represent the projected coordinates of the particle track.
Using the information of the drift time ∆ti = ti − t0, with ti time of detection
and t0 trigger time, the y-coordinate of the track can be calculated. Thus, a 3D
track of the traversing particle can be reconstructed. To minimize the material
budget, the MAGIX TPC is designed with a novel open field cage concept. For
more information please refer to the doctoral thesis of P.Gülker [Gül21].
y
Electric Field
x
z
t1 = t0 + ∆t1
tn = t0 + ∆tn
t0: Trigger
Figure 4.9: Schematic principle of a TPC. The green block illustrates the tra-
jectory of an electron-ion pair. The ion drifts along the homogeneous electric
field to the cathode while the electron drifts to the anode. Each electron gen-
erates a signal after the time ti = t0 + ∆ti with trigger time t0 and drift time
∆ti. The electrons are amplified in each GEM layer. The pad plane anode
can be used to read the signal and determine the x and z coordinate. The drift
time corresponds to the y coordinate.
61
Drift-Cathode
Particle Track
GEM 1
GEM 2
GEM 3
Pad Plane-Anode
Chapter 4 - Accelerator and Experimental Setup
Trigger Veto System
The trigger veto system of each sprectrometer (see Figs. 4.10 and 4.12a) consists
of one layer of segmented plastic scintillator detectors (trigger layer) followed by
several layers of additional scintillation detectors (veto layers) with lead absorbers
in between. The trigger layer at the top of the system is used for fast time and
energy loss measurements. Through its segmentation, it provides the basic hit and
position measurement used to trigger the readout of the TPC. The cross section
of the segments of the trigger layer is diamond-shaped at an angle of 60°. This
results in an overlap in the electron direction. The maximum track divergence
for the scattered electrons is 10°, therefore it is ensured that every particle in the
trigger layer is detected (see Fig. 4.10) [Ste21a; Ste21b]
Figure 4.10: The current 3D concept of the trigger veto system. The cross
sections of the segments of the trigger layer are diamond-shaped.
Figure 4.11: Schematic view of the cross section of the diamond-shaped seg-
mented trigger layer.
Because of the special upside down design of the spectrometers, a more complex
trigger veto system is required. The challenge is to separate the scattered electrons
and cosmic muons as they point in the same direction. Beam-induced neutrons
can also produce signals in the scintillating layers. All of these particles leave
characteristic signatures in the trigger veto system (see Figs. 4.12b to 4.12d) that
allow them to be distinguished. The electrons generate signals in the trigger layer
T and the first veto layer V1. The electrons and electron-induced electromagnetic
showers are stopped in the absorber which extremely reduces the probability of
obtaining an electron-induced signal in veto layers 2 and 3. Unlike electrons,
muons pass through the absorbers and thus generate a signal in the scintillator
62
4.2 Mainz Gas Injection Target Experiment
layers V2 and V3. Therfore, we can distinguish between electrons and muons
using the Signal V2&V3 as a muon veto. The beam-induced neutrons do not
have a specific direction, they usually produce a signal in only one scintillator
layer. The probability that a neutron will produce a signal in both layers T and
V1, or that a random coincidence will occur, is low [Ste21a; Ste21b].
Muon Electron Electron
Trigger T Trigger T
Veto V1 Veto V1
Lead Shield Lead Shield
. .
Neutron . .
. .
. .
. .
. .
. .
Veto V2 Veto V2
Lead Shield Lead Shield
Veto V3 Veto V3
Electromagnetic Showers Electromagnetic Showers
(a) Veto Trigger System (b) Electron Signature
Muon
Trigger T Trigger T
Veto V1 Veto V1
Lead Shield Lead Shield
. .
. Neutron .
. .
. .
. .
. .
. .
Veto V2 Veto V2
Lead Shield Lead Shield
Veto V3 Veto V3
(c) Muon Signature (d) Neutron Signature
Figure 4.12: Schematic diagram of the MAGIX trigger veto system [Ste21a].
Figure (a) shows the detector setup with all types of particles passing through
the detector system. Figures (b)-(d) show the different detector signatures for
the different types of particles expected at MAGIX [Ste21a].
4.2.3 Silicon Strip Recoil Detector Array
For some of the reactions that will be studied at MAGIX it is necessary to de-
tect the recoil particles as well as the scattered electrons. In order to detect the
various low energy heavy charged recoil particles, such as protons, deuterons or
α-particles, MAGIX needs a further detector system in addition to the spectrom-
eters. For this purpose, a silicon strip detector array will be used. It consists of
several silicon strip detectors mounted inside the scattering chamber at a distance
of about 30 cm from the interaction point. Each individual silicon strip detec-
tor can be rotated independently around a common pivot point [Gei21a; Gei21b;
Gei20].
63
Chapter 4 - Accelerator and Experimental Setup
Figure 4.13: Prototype of the silicon strip detector inside a test chamber.
The cooling element is attached to the back of the X1 detector. The readout
electronics is connected to the bottom of the detector. The APV25-S1 front-
end chip is located at the front of the electronics [Gei20].
The detector array is in development. Currently, the X1 silicon strip detector
from Micron Semiconductor Limited is being tested. The active area of the de-
tector is 50× 50 mm2 and consists of a 995 µm thick silicon substrate. On the
back side, the substrate is covered with an aluminum layer. 16 p-doped silicon
stripes with a width of b = 3 mm are embedded in the n-doped substrate. Those
have no entrance window, so, the low-energy charged heavy particles do not lose
energy before depositing their energy in the substrate. The X1 is mounted on
a holder which contains a cooling element, actively cooled by circulating ethanol
(see Fig. 4.13). Since the dark current of the silicon strip detector is extremly
temperature dependent, the signal to noise ratio will be optimized by cooling the
detector down to −20 ◦C. So, for example, α-particles with a kinetic energy of
300 keV can be detected. The readout of the detectors is based on the APV25-S1
frontend chip [Gei21a; Gei20].
The silicon strip detector is a semiconductor detector. Its working principle is
illustrated in Fig. 4.14. In addition to the classical pn-junction a further, more
heavily n-doped layer, called n+, is added to semiconductor detectors. This en-
larges the space-charge region, which forms the sensitive area of the detector. The
detector is operated via external voltage in the reverse direction. If a charged par-
ticle hits the space charge region of the detector, electron-hole pairs are produced.
The holes drift along the external applied field to the stripes where the position
will be detected representativly for the incoming particle. With the APV, the
generated signal charge can be read out on both sides of each stripe. Due to
the charge asymmetry, the y-position along the stripes can be determined. The
64
4.2 Mainz Gas Injection Target Experiment
-
b p
p+
z
n
y
+
x n
Al
(a) Cross section of the silicon strip detector across the stripes. The
16 p-doped stripes have a width b and a distance p to each other.
The spatial resolution in x-direction is σx = √b12 .
Q l Qup−Qdowndown y = 2 · Q +Q Qup down up
l
p+
z
n
y
n+x
Al
(b) Cross section of the silicon strip detector along the stripes. The
produced signal charge is measured for each stripe at both ends.
Due to the charge asymmetry, the position along the y-direction
can be determined.
Figure 4.14: Schematic representation of the working principle of the silicon
strip detector. When a charged particle hits the space charge region of the sub-
strate, electron-hole pairs are produced. The electron drifts along the applied
external field to the anode, whereas the hole drifts to the stripes. [Gei20].
segmentation into stripes of equal width b results in a discrete resolution in the
x-direction. The spatial resolution is σ = √bx 12 . To start data acquisition, the
APV needs a trigger signal. Since most particles are stopped in the detector, no
additional detector can be used to generate the required trigger signal. A novel
frontend-board has been developed by the MAGIX group to obtain this timing
information from the electrons that are generated simultaneously with the holes
and drift to the anode. The board amplifies the electrons and passes them to the
APV as a trigger signal [Gei20; Gei21a].
4.2.4 Zero Degree Tagger – FORE
In a later phase of the MAGIX program, another electron detector, the beam-line
integrated Zero Degree Tagger (ZDT), will be available. This detector is designed
to measure electrons scattered at an angle of less than 0.5° in the forward direction.
The planning of this detector has begun and in this section we will only get a basic
65
Chapter 4 - Accelerator and Experimental Setup
idea of this detector. The project name of this detector is FORE, like the nautical
term for forward, analogous to the two spectrometers.
The idea is to use the first dipole magnet in the beam-line after MAGIX to focus
the scattered electrons based on their energy on a plane. A detector system
located in this area, not yet further defined, will then measure these electrons, in
particular to determine their energy (see Fig. 4.15). For this purpose a special
design of the dipole magnet is necessary. To ensure that all the requirements in
the planning process are taken into account the MAGIX group is working closely
with the MESA accelerator group. According to current plans, the magnet will
have a deflection angle of 45° to guide the unscattered electrons back to the
cryomodules [Led20; Sim21b]. Since the magnetic field cannot be adjusted as
desired, the measuring range of this detector is restricted by a fixed reaction and
beam energy. However, by reducing the beam energy, different energy ranges
can be measured. In inelastic reactions, the energy of the scattered electrons is
significantly lower than the beam energy. This makes it possible to separate the
scattered electrons from the beam and detect them in the ZDT. For example, in
the reaction 16O(e, e′α)12C, to split 16O into 12C and α, the energy transfer ω must
be greater than 7.16MeV (see Table B.5). The electrons involved in this reaction
therefore lose at least 7.16MeV and can thus be separated from the beam and
measured in the ZDT.
Figure 4.15: Schematic representation of the zero degree tagger for MAGIX.
The first deflection magnet is used to focus the electrons onto a plane where a
detector system will measure them. The red area is to represent a scattering
plane to show the area of the ZDT.
4.3 S-Factor Measurement at MAGIX
A general overview of the electro-disintegration experiment is given in Section 2.3.2.
In this section, we will discuss the planned measurement of the electro-disintegra-
tion reaction 16O(e, e′α)12C at MAGIX. First, we will discuss the general setup
66
4.3 S-Factor Measurement at MAGIX
and show how the various challenges of this measurement will be solved. This
particular experiment will be performed in three stages. At the end of this sec-
tion, these different stages, their improvements, and their planned measurement
ranges will be discussed.
4.3.1 Discussion of Electro-Disintegration Experiment
To determine the S-factor of the reaction 12C(α, γ)16O, MAGIX will measure
the cross section of the electro-disintegration reaction 16O(e, e′α)12C. Therefore,
electrons will scatter on an oxygen gas jet. The scattered electrons will either
be measured at the MAGIX spectrometers or at the zero degree tagger in co-
incidence with the recoiled α-particle. The α-particle will be detected with the
silicon strip detector array. The four-momentum of the virtual photons qµ and
the center-of-mass energy W will be calculated by measuring the four-momentum
of the scattered electron. With the measured count rate Ṅ the cross section of
the electro-disintgration reaction Eq. (3.39) can be determined using the formula
[Pov+15]
Ṅ = σ · L,
where the quantity L represents the luminosity. The luminosity describes the
number of particle collisions per area per time. For a linear accelerator the lumi-
nosity can be calculated using the equation [Pov+15]
L = Ṅa · ρb,
with the number of incoming beam particles per time Ṅa and the areal particle
density of the target ρb. For an electron accelerator we have Ṅ = Ia with thee
beam current I and the elementary charge e. The areal particle density of the
oxygen gas jet will be about 2× 1018 atoms/cm2. So, with a beam current of 1mA
the luminosity is around 1.25× 1034 cm−2 s−1. Since both the jet current and the
gas density are not constant, the luminosity is monitored in the final experiment.
One problem of the electro-disintegration technique is the contamination of the
oxygen gas with 17O, 18O and 14N. As discussed in [FDM19] the cross sections of
the reaction 17O(e, e′α)13C, 18O(e, e′α)14C, and 14N(e, e′p)13C are several orders
of magnitude larger than the cross section of the reaction 16O(e, e′α)12C close to
the Gamow peak. However, Friščić, Donnelly, and Milner also showed that by
measuring the kinetic energy of the recoil particles and using the Time of Flight
(ToF) information, the particles can be assigned to the corresponding reactions.
Thus, an appropriate filter can be used in the analysis to eliminate the back-
ground created by this contamination. For this purpose, a minimum distance of
30 cm between the silicon strip detectors and the interaction point is recommended
[FDM19].
One advantage in the MAGIX setup is, that neither the electrons nor the α-
particles pass any windows before they are detected by their individual detector.
This allows to calculate the cms energy and the angular distribution very accu-
rately. With the silicon strip detector array, we can measure the cross section for
many different angles of the α-particles. This allows us to determine the longitu-
dinal, transverse, and interference terms and, in particular, the multipolarities E1
and E2 very accurately. The ground-state transition cross sections σE1 and σE2 are
then analyzed using the R-matrix method and the S-factor at the Gamow peak
is extrapolated. Due to its high luminosity, low background and high accuracy in
determining the parameters, MAGIX is an excellent experiment for determining
the S-factor close to the Gamow peak.
67
Chapter 4 - Accelerator and Experimental Setup
4.3.2 Discussion of the Planned Extension Stages
The experiment is performed in three successive phases. The phases are numbered,
starting with number zero. In Phase 0, the MAGIX target is used at Mainz
Microtron (MAMI) in Mainz, Germany. In this phase, the target system and
the silicon strip detector array are tested. In addition, this phase will be used
to verify the theory. Phase 1 will take place when MAGIX is commissioned at
MESA. After some further testing, the S-factor measurement will be performed.
Phase 2 will begin after MAGIX is expanded by the ZDT. At each stage, the aim
is to improve the already achieved results. In the following, we will discuss these
phases in detail.
Phase 0: MAGIX@MAMI
In Phase 0 we will use the MAGIX target adapted to the A1-experiment at the
existing accelerator MAMI in Mainz, Germany. The electron beam will have an
energy of E0 = 195 MeV and a current of I = 100 µA. The scattered electrons
will be detected by one of the A1-spectrometers and the recoiled α-particle will
be detected by several silicon strip detectors. These will be mounted around the
interaction point at a fixed angle and a distance of around 30 cm (see Fig. 4.16).
The arrangement of the silicon strip detectors will be optimized to measure the
E1 ground state transition, as discussed in Section 2.3.1. Since the scattering
chamber at A1 is smaller than the final one at MAGIX, only a few silicon strip
detectors can be installed in the chamber. Preparations for this phase have only
just begun. According to current planning, the phase is to be carried out between
2022 and 2023.
In this phase, MAGIX will determine the S-factor for cms energies in the well know
region of 1.7MeV to 2.7MeV. The goal of this phase is to test the crucial parts of
the setup as well as the reconstruction and analysis of the data. The results will
be compared to the well-known existing data in this parameter range. Thus, we
can test the theory of electro-disintegration and make a first contribution to the
global S-factor data. In addition, we can determine the longitudinal as well as the
two interference terms in this paramter range. This will allow us to improve the
simulation and optimize the setup for the α-particle detectors in the next phases.
Phase 1: MAGIX@MESA
The Phase 1 can begin after MAGIX is commissioned at MESA. The electron
beam will have an energy of E0 = 25− 105 MeV with a current of I = 1 mA. We
will detect the scattered electrons with the high resolution spectrometers and the
α-particles with the full size silicon strip detector array. The spectrometers will be
placed at the lowest possible angle of around 15°. Each silicon strip detector can
individually rotate around the interaction point. Thus, we can choose different
angular arrangements to optimize the determination of the E1 and E2 ground
state transition. An example of the experimental setup can be found in Fig. 4.16.
With this setup, MAGIX will be able to perform the multipole expansion of the
cross section of the reaction 16O(γ, α)12C to determine the S-factor for the E1 and
E2 contributions. According to current planning, the Phase 1 is to be carried out
between 2023 and 2026.
The goal of this phase is to determine the S-factor in the cms energy range above
0.9 MeV with high precision. For this purpose, we will determine the longitudinal,
transverse, and both interference terms in the known range above 0.9 MeV with
68
4.3 S-Factor Measurement at MAGIX
Spectrometer
Silicon Strip
Detector Array
Interaction Point
Electron Beam
Figure 4.16: Schematic view of the experimental setup for Phase 0 and Phase
1. In both phases the scattered electrons will be detected by a spectrometer
while the α-particles will be detected by the silicon strip detector array. The
single silicon strip detectors will be mounted inside the scattering chamber
around the interaction point.
high accuracy. In addition, we will improve the analysis software and also include
a Monte Carlo simulation to extrapolate the S-factor to the Gamow peak and
determine the errors. For data analysis, we will use the R-matrix method. This
phase will also be used to calibrate the detectors and prepare for the Phase 2.
Phase 2: MAGIX@MESA including Zero Degree Tagger
In Phase 2 we will detect the electrons with the ZDT instead of using the spec-
trometers. The electron beam will have an energy of E0 = 25 − 105 MeV with
a current of I = 1 mA. The cross section Eq. (3.39) is proportional to the Mott
cross section, which can be written using Extr(em)e Relativistic Limit (ERL) as
α2 cos2 (θ= 2σMott ) .
4E2 sin4 θe 2
To increase the measured cross section there are two possible parameters we
can change in the experimental setup. Decreasing the beam energy is limited
to E0 = 10 MeV and at those energies the cross section will increase by a factor
of about 100. If we use the beam integrated zero degree tagger to detect the
scattered electron with scattering angles below 0.5° we can further increase the
measured cross section with a factor of around 106. But we have to keep in mind,
that in this parameter range the ERL cannot be used. Therefore, the cross sec-
tion and all included quantities need to be used in general form as derived in
69
Chapter 4 - Accelerator and Experimental Setup
Chapter 3. The associated analysis and also the simulation must be adapted ac-
cordingly. However, a major advantage at this point is that for such small angles
of the scattered electrons, it is not necessary to measure the angle and we can
integrate directly over the angular acceptance of the ZDT. According to current
planning, the Phase 2 will be carried out between 2026 and 2032.
Since the electrons are scattered with an angle close to 0° the carbon-α system
is boosted nearly in forward direction. This simplifies the experimental setup,
since the silicon strip detectors can be arranged symmetrically on both sides of
the beamline. Thus, also the determination of the multipolarities E1 and E2 is
simplified. With this experimental setup MAGIX plans to determine the S-factor
in the so far unknown cms energy area below 0.9MeV. However, the cross section
of the 12C(α, γ)16O decreases by one order of magnitude per 0.1MeV step for cms
energies below 1MeV. Even though the cross section is roughly 106 larger than
the cross section of Phase 1 it is still challenging to determine the S-factor at the
Gamow peak. Another advantage is that the longitudinal and both interference
terms are very small for small Q2, so the transverse part can be extracted more
easily. In order to determine the possible parameter range that MAGIX can access
to determine the S-factor of the reaction 12C(α, γ)16O, a Monte Carlo simulation
was performed. The results of this simulation for phases 1 and 2 are discussed in
Chapter 6.
Zero Degree Tagger
Silicon Strip
Detector Array
Interaction Point
Electron Beam
Figure 4.17: Schematic view of the experimental setup for Phase 2. The scat-
tered electrons will be detected by the zero degree tagger while the α-particles
will be detected by the silicon strip detector array inside of the scattering cham-
ber.
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CHAPTER 5
Development of the MAGIX Slow
Control System
As mentioned in the previous chapter, some of the MAGIX components exist and
initial tests are being performed in different environments. For example, the cur-
rent TPC and one silicon strip detector are being tested in their respective test
stands. Since a common control software is needed for all these components, the
development of a MAGIX Slow Control System (MXSlowControl) was necessary.
One goal of this thesis was the development of a basic version of the MXSlow-
Control system which will be used for MAGIX at MESA. The development is in
progress as new devices constantly will be added to the experimental setup. So
far, most of the components used by MAGIX are controlled by MXSlowControl.
In this chapter we will discuss major parts of the MXSlowControl system, starting
with the basic requirements for such a system. Then we will review the base
software for MXSlowControl: EPICS. After a review of the community modules,
an introduction to the development of MXSlowControl will follow, in particular
to the modules created in the context of this thesis. At the end of the chapter,
an example of the setup for a test stand for the silicon strip detector is given.
5.1 Requirements for a Slow Control System
Before looking at the requirements for the MAGIX slow control system it is nec-
essary to understand the difference between a slow control software and a data
acquisition for an experiment. Naively, you can separate both terms by looking
at the data rate and the amount of data. The data acquisition rate of the detec-
tors used by MAGIX is in the range of several kHz or MHz [Sch21]. This means
that a high amount of data must be written to a hard disk in a short amount
of time. In contrast, it is not necessary to read or write control parameters such
as supplemental voltage, pressure, flow rate, etc. at such high rates. In a run-
ning measurement, most of these parameters remain more or less constant or are
accessed only in specific time rates in the order of a few Hz. The associated com-
munication effort and also the amount of data for slow control is therefore several
orders of magnitude smaller than for data acquisition.
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Chapter 5 - Development of the MAGIX Slow Control System
Based on the over 40 years of experience with the MAMI accelerator and ex-
periments performed there, we are able to prepare a list of requirements for the
MAGIX slow control. First of all, it is necessary to separate the system, and thus
the software, into smaller parts. Each of these parts have to run independently
and control its own devices. Communication between these decentralized parts
and with a Graphical User Interface (GUI) is necessary. This ensures the stabil-
ity of the system, as tasks can be distributed among several computers and the
failure of one subsystem does not directly affect the entire system. In addition, it
is desirable that the slow control systems of the accelerator and the experiments
can communicate with each other. For example, the system of the experiments
should be able to monitor and archive relevant accelerator data.
Furthermore, easy integration of new devices and replacement of damaged devices
must be possible without large programming effort. To control all parameters one
GUI for the entire control system is needed. However, instead of having only one
access to the system, both, the slow control and the GUI, must be accessible from
the whole experimental area.
The slow control system must allow the setting of various system states, especially
the configuration of connected devices. There must also be a secure alarm system
that allows automatic response to time-critical errors. In order to perform highly
precise experiments, all parameters must also be properly archived so that the
analysis can reproduce the system state of the available data at any time.
Although almost all devices are delivered with their own software, this is not used
in the final experiment. Even with a small number of different devices, the con-
trol and readout of the devices becomes confusing. In addition, communication
between the devices is usually not possible, which would pose a risk to the exper-
iment. For example, if the pressure in the scattering chamber increases too much,
the system must be able to close the corresponding valves automatically.
One possibility is to implement a custom-made slow control. This has the ad-
vantage that it can be perfectly tailored to the needs of the experiment. The
disadvantage is that it is then the responsibility of the developer to ensure the
long-term support. Alternatively, an already existing and well tested slow control
system can be used. One of these is EPICS. As part of a bachelor thesis, a feasi-
bility study was performed to investigate the advantages of an EPICS based slow
control system for MAGIX [Emi17]. Based on the results of the investigation and
the fact that both P2 and MESA will use an EPICS based slow control system,
it was decided to also use EPICS as basis.
5.2 Experimental Physics and Industrial Control System
Experimental Physics and Industrial Control System (EPICS) is a decentralized,
slow control system. This means that the servers and clients involved run inde-
pendently of each other and communicate via a common network. It is a control
system architecture supported by a huge international collaboration. It was de-
veloped by the collaboration of the Argonne National Laboratory, the Los Alamos
National Laboratory, and other Departments of Energy Laboratories [EPICS21;
APS04]. Due to the modular concept of EPICS, many groups worldwide work on
different software toolkits and make them available to other users via the commu-
nity. These can easily be installed to extend the base EPICS version. In addition
to the software modules, many device drivers are also offered on the EPICS home-
page, with which a device can directly be integrated into EPICS. Furthermore,
72
5.2 Experimental Physics and Industrial Control System
some manufacturers already provide their own EPICS drivers for their devices.
This allows EPICS to be adapted very flexibly to different experiments.
In this section, we will discuss the principle of the EPICS-BASE system and
how the various components work. We start with the Input-Output-Controller
(IOC), which is the main program of EPICS. It handles all communication with
the connected devices and provides all control parameters over the network. The
core of the network communication is the communication protocol Channel Access
(CA). Next, we discuss the CA client (CAC), which can write, read, and monitor
data from the IOCs. At the end of this section, we will get a detailed explanation
on how the channel access communication protocol works. For more information
about EPICS please refer to [EPICS21; APS04].
5.2.1 Input-Output-Controller
The main process of EPICS is the Input-Output-Controller (IOC). Each IOC
can be divided into three different major software components, the CA server
(CAS), the database (DB) and the Device Support (see Fig. 5.1) [B3.14; APS04].
Processes run in the background to perform the individual tasks of the three
components and enable the communication between them. After executing the
IOC process the IOC shell opens. In the IOC shell the Database and Device
Support needs to be configured before the CA server can be started.
Network
IOC
Channel Access Server
Database
Device/Driver Support I/O Hardware
Figure 5.1: Overview of the major sotware components of an IOC. The CA
server provides process variables over the network. The database is a collec-
tion of EPICS records to collect, process and distribute data. The device or
driver support is the interface between the database and the hardware. [APS04;
B3.14].
Database
The database is a purely IOC-internal software module that, as the name suggests,
contains the data objects. The data objects are called records in EPICS. Each
record is of a specific type and can be identified by its unique name. Typical types
are [B3.14]
• analog input (ai): used for reading double values from devices
• analog output (ao): used for writing double values to devices
• long input (longin): used for reading long values from devices
• long output (longout): used for writing long values to devices
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Chapter 5 - Development of the MAGIX Slow Control System
• bool input (bi): used for reading bool values from devices
• bool output (bo): used for writing bool values to devices
• calculation record (calc): calculates a value from one or multiple other
records
• calc out (calcout): calculates a value from one or multiple other records and
writes this value to a device
Depending on the type, records have different properties, called fields. How a
record type is structured and what tasks it can perform is specified in the database
definition file (DBD). EPICS base contains a number of predefined types. These
can be modified, or completely new types can be created. Typical record fields
are [B3.14]
• NAME: Unique name of the record
• DESC: Contains a human readable description of the record
• TIME: Timestamp of the record
• VAL: Value of the record
• EGU: Engineering units of the value
• SCAN: Periodic time interval to automatically process the record
• INP: Input specification for Device Support
• OUT: Output specification for Device Support
Each record is an active object. The tasks provided are reading or writing data
to other records or devices, checking the range of values, issuing a warning, and
so on [APS04]. The scope of the tasks depends on the type of the record. For
more information on record types and the assigned fields please refer to the Record
Reference Manual at [B3.14].
The database is configured via different text files, with the format ending .db. It
is common to create a separate db file for different devices. The db files can be
loaded with different patterns. Thus, one db file is sufficient for identical devices.
These can then simply be loaded with different parameters. There are multiple
IOC commands to load a specific db file. In code 5.1 an example is given how a db
file can be loaded with different patterns in the IOC shell. Further possibilities for
loading db files will be discussed in Section 5.5.8. Code 5.2 shows an extract from
the devMPT200.db file. Here a calc record named $(P)$(NR)Pressure is defined
and some properties (fields) are set. The CALC field specifies how the raw value
is converted to mbar. With the patterns of code 5.1 the calc record will be loaded
two times with the unique names MPT200:0:Pressure and MPT200:1:Pressure.
Code 5.1: IOC shell example. The db file "devMPT200.db" is loaded with
different patterns using the IOC shell command dbLoadRecords.
# Load database for two MPT200 with different configuration pattern
dbLoadRecords("$MXSlowControl/db/devMPT200.db","P=MPT200,␣NR=0,␣BUS=ttyUSB0")
dbLoadRecords("$MXSlowControl/db/devMPT200.db","P=MPT200,␣NR=1,␣BUS=ttyUSB1")
The database is responsible for collecting, processing and distributing the informa-
tion for all loaded records. For this purpose, it distributes these tasks to different
processes depending on the configuration. In addition, it serves as an interface
74
5.2 Experimental Physics and Industrial Control System
Code 5.2: Example of a calc record. The input values are from the record
$(P):$(NR):GetPressureRaw in U_Expo_New format [MPT200]. The CALC
field specifies the conversion from the raw value into mbar. The SCAN field
with the value Passive defines, that the calculation is performed by an external
trigger. In this case the record $(P):$(NR):GetPressureRaw will trigger the
calculation each time its value changes.
record(calc, "$(P):$(NR):Pressure")
{
field(DESC, "Pressure␣in␣mbar;␣device␣$(BUS)")
field(INPA, "$(P):$(NR):GetPressureRaw")
field(CALC, "FLOOR(A/100)∗10^(A%100−23)")
field(SCAN, "Passive")
field(EGU, "mbar")
field(PREC, 7)
field(HOPR, 2e+3)
field(LOPR, 1e−5)
}
and toolkit for creating an individually configured IOC that performs the required
tasks. The database enables the other software modules to access all process in-
formation. To read and write data from devices, the database (records) makes
the appropriate requests to device support [B3.14; APS04].
The Device Support Interface
Device support, also called driver support, is the interface between the database
and the input/output hardware. Any device that can be connected via USB, se-
rial port, network, etc. can be addressed with the device support. To add new
devices to EPICS, the driver for that device must be written in C or C++. This
driver can be loaded into EPICS using the device support interface. Code 5.3
shows an example of such an interface. The C++ struct devSIDET_ai is ex-
ported to EPICS and can be accessed with the device support module. In this
example two functions init_recordSIDET and read_aiSIDET have to be defined.
The init_recordSIDET function is executed by each record associated with the
SIDET device. This is performed during the initialization of the database prior
to any other processes getting started. This allows loading and configuring the
device driver. After all records are initialized, the database starts collecting
data. Thereby the device support executes the function read_aiSIDET every
time a record requests data from the device. In addition to the device driver, the
init_recordSIDET and read_aiSIDET functions must also be written in C or C++
by the user. A major advantage of using EPICS is that the EPICS community
provides some modules to simplify the integration of new devices. In addition, a
large number of device supports for various devices are already available on the
EPICS home page [EPICS21]. Those can easily be installed and used to control
new devices. The communication structure for devices is often the same, thus a
new device can be included by adjusting the device support appropriately [B3.14;
EPICS21; APS04].
CA Server
The CA server (CAS) is the interface between the database and the network.
The record names are combined with its associated field names to form an indi-
vidual identification string called Process Variable (PV). For example, with the
configuration code 5.1 and the record code 5.2, the PVs MPT200:0:Pressure,
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Chapter 5 - Development of the MAGIX Slow Control System
Code 5.3: Example of a Device Support C interface. The C++ struct
devSIDET_ai is exported, so that EPICS can access it. The function
init_recordSIDET is executed from the database during the initialization of
the record. To read a value from the device the function read_aiSIDET is
executed from the Device Support [B3.14].
1 #inc lude " devSup . h "
2 #inc lude " aiRecord . h "
3 #inc lude " ep icsExport . h "
4
5 extern long init_recordSIDET ( s t r u c t aiRecord ∗ pai ) ;
6 extern long read_aiSIDET( s t r u c t aiRecord ∗ pai ) ;
7
8 s t r u c t {
9 long number ;
10 DEVSUPFUN repor t ;
11 DEVSUPFUN i n i t ;
12 DEVSUPFUN in i t_reco rd ;
13 DEVSUPFUN get_io in t_in fo ;
14 DEVSUPFUN read ;
15 DEVSUPFUN spec i a l_ l i n conv ;
16 } devSIDET_ai = {
17 6 ,
18 NULL,
19 NULL,
20 init_recordSIDET ,
21 NULL,
22 read_aiSIDET ,
23 NULL
24 } ;
25
26 epicsExportAddress ( dset , devSIDET_ai ) ;
MPT200:0:Pressure.DESC, MPT200:0:Pressure.EGU, . . . are created. The CA
server provides access to all PVs via the network. However, this implies that a
record name may not be assigned twice within the same subnet [B3.14].
5.2.2 CA Client
A CA client (CAC) is a program that accesses the process variables of a CA server.
A client may connect to several CASs to write, read, or monitor data from PVs.
Since it may be necessary for a record to request information from records loaded
for processing in another IOC, each IOC may also contain a CA client. EPICS
base is supplied with three different CAC programs [B3.14]:
caput: write a value to one PV. Example: caput PV0 4.2, sets the value of PV0
to 4.2.
caget: read a value of one PV. Example: caget PV0, returns the value of PV0.
camonitor: monitors the values of one or more PVs. Whenever the value of
one of the PVs changes, the new value is sent to the CA client camonitor
together with a timestamp. Example camonitor PV0 PV1 PV2, monitors
the value of PVs PV0, PV1, and PV2.
In addition, each GUI like CSS or archiver is a CA client application. EPICS also
provides libraries to access the channel access and thus the PVs of the IOC via
Python scripts or C++ programs. All these programs are then also considered as
CA client processes. The application fields of a CA client can be designed very
76
5.2 Experimental Physics and Industrial Control System
flexibly and adapted to the requirements. Figure 5.2 shows an overview of the
task distribution of the IOCs and CA clients in an experimental setup.
IOC CA Client
IOC CA Client
IOC CA Client
Channel Access
Network
IOC
Channel Access Server Channel Access Client Network Attached
provides Process Variables (PVs) • Write, read and monitor data of PVs of
a CA server (IOC) Devices
• Can connect to several CA servers
Database • Client program examples
• Application specific
– caput, caget and camonitor
• Collection of EPICS records of
various types – GUI (e. g. CS-Studio)
– Archiver
• Collect, process and distribute – Python, C++data , . . .
Device/Driver Support
Interface between Database and I/O Hardware
Hardware
Figure 5.2: Overview of the task distribution for the Input-Output-Controller
(IOC) and the CA clients (CACs). An IOC always contains a CA server, but
can also include CA client processes.
5.2.3 The Channel Access Communication Protocol
Channel Access (CA) is the communication protocol of EPICS. This enables the
CA clients to communicate with the CA servers. Channel access is based on the
two network protocols User Datagram Protocol (UDP) and Transmission Control
Protocol (TCP). Figure 5.3 shows an overview of the CA’s working principle. If
a CA client application wants to access a process variable, the IOC providing
this information must first be determined. The CA client sends a UDP broadcast
request to the subnet asking which IOC contains a particular PV. The correspond-
ing IOC responds via UDP. The client then opens a TCP connection to the IOC
which is used for further communication. If only one value is requested or set, as
in the case of caget or caput, the TCP connection is subsequently terminated.
In the case of a monitoring request, as from camonitor, the connection is held.
The IOC creates a callback function for each monitored PV, which forwards the
value change of a PV directly to the client(s) [B3.14].
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Chapter 5 - Development of the MAGIX Slow Control System
1. UDP Broadcast Request:
Python
Archiver CSS Who provides the PV "PV1"?Script 2. UDP Reply: I do!
CA Client CA Client CA Client
3. TCP Connection: Let’s talk!
Channel Access
Network
CA Server CA Server CA Client CA Server CA Client Network Attached
IOC 1 IOC 2 IOC 3 Devices
PV1 PVs PVs
PV2
PV3
. . .
Figure 5.3: Overview of the channel access process. A client starts a search
request for each PV, to find its server via UDP. After identifying the CA
Server the following communication is done via TCP.
5.3 Community Modules
As mentioned earlier, there is a long list of modules provided by the EPICS
collaboration and community to add further functionalities to the base version of
EPICS. In this section, we will discuss some of these modules and highlight the
benefits and, to some extent, the need for them. We will limit our discussion to
the modules that are necessary for MXSlowControl.
5.3.1 The Asyn Module
Most devices can be controlled via low-level communication. This means that
only a socket has to be configured. By using simple byte streams the control
commands are written to the device via the socket. The devices process the
commands and eventually send a feedback to the socket. This is where the asyn-
Driver module comes into place. It provides a low-level driver interface that han-
dles the communication with the devices. For this purpose IOC shell commands
are implemented to configure the communication port of the devices. Code 5.4
shows an example of such a configuration. In addition, a set of asyn records is
created to provide generic access to a device port. The asynDriver module han-
dles all defined port accesses. For diagnostics, asyn provides a logging function
that allows the output of error messages down to the entire byte stream, called
trace [ASYN21]. Since communication is handled by asynDriver, this simplifies
the implementation of new devices. Users only need to include the device-specific
code in the device support and specify how the response is processed.
Code 5.4: Example for an asynDriver configuration of a serial port. The device
/dev/MPT200 is loaded with asynDriver and stored under the port name L0.
Then the baud rate, data word length, parity and stop bit are configured for
this port.
# Set up ASYN ports
# drvAsynSerialPortConfigure port ipInfo priority noAutoconnect noProcessEos
drvAsynSerialPortConfigure("L0","/dev/MPT200",0,0,0)
asynSetOption("L0", −1, "baud", "9600")
asynSetOption("L0", −1, "bits", "8")
asynSetOption("L0", −1, "parity", "no")
asynSetOption("L0", −1, "stop", "1")
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5.3 Community Modules
5.3.2 The StreamDevice Module
A supplement for asynDriver is the StreamDevice module, which is a generic device
support that implements a string stream communication interface. StreamDevice
allows to include devices without recompiling the slow control system. For this
purpose, StreamDevice is configured with so called protocol files that describe
the communication with a device. The protocol files include one protocol for
each function of the device, where the device-specific commands are defined in
ASCII code. The string stream can be formatted and values can be passed using
format converters similar to the C functions printf and scanf. An example of a
protocol file and the corresponding record can be found in code 5.5. StreamDevice
can be used for all devices that provide a simple string stream communication.
To adapt StreamDevice to specific needs, it can be expanded with further format
converters. If a device can not be addressed with asynDriver, StreamDevice can
also be extended with further bus drivers via the StreamDevice BUS Application
Programming Interface [SDev21; SDev2.8].
Code 5.5: Example of a protocol and the corresponding record. The record file
devMPT200.db is loaded into the IOC memory with the given pattern. As in
code 5.4 defined LO is refered to the serial port of the device MPT200. The record
field DTYP with the value stream defines that the record loads the StreamDevice
driver support. The string of the field INP is passed to StreamDevice to config-
ure the communication. This specifies that the command reading the pressure
from the device connected to port $PORT is defined in the protocol GetPressure
in the protocol file devMPT200.proto. The number 001 is passed as physical
address to the protocol as an argument. The format of the control command
out and the response in are defined in the protocol. The value is read in 6
digits decimal U_Expo_New format
IOC shell command
dbLoadRecords("devMPT200.db","P=MPT200,R=,PORT=L0,ADDR=001")
FILE: devMPT200.db
...
record(ai, "$(P)$(R)GetPressureRaw")
{
field(DESC, "Get␣Pressure␣from␣device␣MPT200␣as␣U_EXPO_NEW")
field(DTYP, "stream")
field(INP, "@devMPT200.proto␣GetPressure('$(ADDR)')␣$(PORT)")
field(SCAN, ".1␣second")
}
...
FILE: devMPT200.proto
...
CMDREAD = "740"; # Pressure in [hPa], readable, size=6, u_expo_new "100023"
# Remote Control Command Format:
# Adress, Action, Command, Size, Buffer, Checksum, Terminator
GetPressure {
# Send Control Command
out $1, "00", $CMDREAD, "02", "=?", "%+<sum8>", "\r";
# Receive Data Response
in $1, "10", $CMDREAD, "06", "%6d", "%+<sum8>", "\r";
}
...
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Chapter 5 - Development of the MAGIX Slow Control System
The advantage of StreamDevice in combination with asynDriver is that any device
that can be controlled via a string stream can be integrated into EPICS without
any programming effort. Since many devices support this, the integration of most
devices is simplified by these two modules. Only the protocols for the configura-
tion of StreamDevice and the asyn port have to be defined for each new device.
However, this requires minimal programming skills, so even non-expert users can
integrate and control devices in EPICS. With the inclusion of StreamDevice and
asynDriver in EPICS, this point of the requirements is therefore also fulfilled.
5.3.3 The Autosave Module
Each time an IOC is restarted, the IOC state is reset to the default configuration
defined in the db files. Even if the input records update their value by the data
request itself, the output records remain in the default configuration. Regardless
of the fact that the IOC does not write any data to the devices when loading
the db files, no statements can be made about the set values of the data sets
after the restart due to this circumstance. Unfortunately, in many cases it is
also not possible to read back the value that was send to the device. This leads
to a problem, because all data of an experiment must be available at any time.
Naively, one way to avoid this problem is not to restart any IOC during the entire
runtime of the experiment. However, this can not be avoided for a number of
reasons. For example, devices may need to be replaced, which may require an
IOC to be restarted. Also, IOCs or the operating system may crash due to an
error and cause a restart.
The Autosave module provides a solution to this problem by adding a function to
preserve the values of an IOC through a restart. Therefore, the autosave module
stores the information of a set of PVs in files like an archiver. In addition, this
module adds the functionality to configure the database before a process starts
its tasks to the IOC. The combination of both functions allows an IOC to load
the latest configuration into the database before resuming work. Autosave only
saves and loads PVs that have been defined by the user. The PVs are specified
in files that can be configured via patterns when loaded into the module, just like
the db files [A5.7].
5.3.4 The PyEpics Library
In some cases it can be very useful to have small programs controlling or moni-
toring multiple PVs simultaneously. This can be a simple BASH script or up to
a powerful C++ program. One programming language that has proven to be very
capable for this task is python. Python is a dynamic scripting language. It allows
object-oriented programming and can be extended by numerous libraries. Python
is characterized by its flexible usability and high user-friendliness due to its clear
syntax. Since the code is interpreted at runtime, it is not necessary to compile
the programs in advance, as is the case with C or C++. This makes it possible to
develop and run programs for various tasks very quickly and easily [Python3].
The module EPICS Channel Access for Python (pyEPICS) builds an interface
between channel access and python. After loading the pyEPICS module to a
python script the command-line tools caget, caput and camonitor can be used in
python. In addition, the module extends python with an object-oriented interface
by providing a PV class. The PV class exposes an EPICS process variable as a
full-featured and easy-to-use python object. Among other aspects, these objects
can be used to create callback functions that monitor values and perform a specific
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5.4 The Graphical User Interface - Control System Studio
action when needed [PyE3]. Due to the simple language of python in conjunction
with CA, any user can create a variety of Python scripts to perform various tasks
related to slow control. Such a script can be used, for example, to switch between
different states of a system. For safety reasons, scripts can also be executed to
monitor specific parameters and automatically set the system to a safe state when
a critical threshold is reached.
5.3.5 The dev-gpio Module
The dev-gpio module can be used for IOCs running on a Raspberry Pi (RPi) or
a BeagleBone to provide easy access to the general purpose input-output (GPIO)
pins. An IOC shell command is used to load the lookup table for the GPIO
names. After that, the voltage of a GPIO can be read or written by a bool record.
Thereby the GPIO pin can be configured as active high or active low [GPIO]. An
example can be found in code 5.6.
Code 5.6: Example of a record with dev-gpio. The IOC shell command
GpioConstConfigure loads the lookup table for the GPIO names of the Rasp-
berry Pi 2 B. The bool-in record is connected to the dev-gpio module via the
devgpio type. The input field is used to specify that the values of GPIO pin 4
are read back as active low. The ZNAM and ONAM fields define a human readable
value associated with the zero and one values.
IOC shell command
GpioConstConfigure( "RASPI␣B+" )
FILE: devValveControl.db
...
record(bi, "$(P):Valve15")
{
field(DTYP, "devgpio")
field(INP, "@GPIO4␣L")
field(SCAN, ".2␣second")
field(ZNAM, "closed")
field(ONAM, "opened")
}
...
5.4 The Graphical User Interface - Control System Studio
There are several solutions for a Graphical User Interface (GUI) for EPICS. One
of them is Control System Studio (CSS), which is a collection of various tools
for controlling and monitoring data for EPICS based control systems. The main
component of CSS is the Operator Interface (OPI). To create a control interface
tools can be dragged and dropped onto an OPI. Each tool can easily be linked
with a PV. The tools range from simple text fields to complex data browsers.
CSS makes it easy to create a graphical user interface for new devices [CSS21].
Figure 5.4 shows an example of a CSS OPI. An additional feature is that scripts,
such as python scripts, can be connected and executed within the OPI.
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Figure 5.4: Example of a CSS OPI. This OPI is used to control the cold head
and the booster stage. Different information like the mass flow or the pressure
at different positions is also shown on this OPI.
5.5 MAGIX Slow Control System
The MAGIX Slow Control System (MXSlowControl) is developed based on the
EPICS framework. EPICS provides a shell command called makeBaseApp to create
a base application for this purpose. It creates a complete directory environment
including an IOC application. The slow control system of the experiment can
be built there, especially the db files and device supports. Like the community
modules, the developed modules are standalone libraries and are installed in the
EPICS base framework. They can be linked to MXSlowControl if required. In this
section some of the most important developments of the MXSlowControl system
of this work are presented.
5.5.1 Installation Script
Installing EPICS, the (community) modules and MXSlowControl on new operat-
ing systems is a complex task. In addition to creating the appropriate environment
variables and integrating the libraries and applications into the system, many con-
figuration files have to be adapted manually to link the individual parts together.
Only when EPICS base and all modules are correctly linked and the corresponding
variables are set up, MXSlowControl can access them. This makes the installa-
tion a time-consuming part and troubleshooting incorrect settings turns out to be
difficult.
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5.5 MAGIX Slow Control System
Since the installation steps are identical for debian-based operating systems, an
installation script was implemented that automatically installs EPICS and the
community modules. The script installs all required packages to the operating
system before it starts installing EPICS. The BASH commands sed, cat and
echo are used to customize the configuration files of EPICS and the modules. If
the script is executed on a RaspberryPi, this is detected and the dev-gpio module
will also be installed. The script is configurable. At the beginning of the script it
can be selected what MAGIX modules have to be installed in addition. The script
creates all definitions of the environment variables of EPICS and the modules in
a file called thisEPICS.sh. This is installed in /etc/profile.d/ so that the
operating system sources it automatically at every reboot. After running the
installation script a full working version of EPICS and the selected modules will
be installed on the operating system. The installation script can be found in the
appendix under code C.1.
Similar to the installation script for EPICS, an installation script for MXSlow-
Control has been implemented. This script builds the MXSlowControl IOC base
application, links all necessary modules to it and copies all files, such as the db
files, into the appropriate directories. Each MAGIX module is made with its own
installation script and can be installed separately if needed. The installation script
of MXSlowControl and one MAGIX module can be found in the appendix under
code C.2 and C.3.
5.5.2 Introduction to Environment Variables
Environment variables on operating system level contain information that is avail-
able for all applications. They are used to adapt applications to the environment
they are running in. Environment variables can be accessed through unique strings
[Deb]. To run EPICS a set of environment variables have to be defined or mod-
ified. As mentioned in the section before, all the definitions are set in the file
thisEPICS.sh. After sourcing this file all definitions will be loaded into the
system. For example, to tell the system where to find the EPICS executable
programs, such as caget, the command locations have to be specified in the en-
vironment variable $PATH. To load libraries, the system must know where to look
for them. Therefore, the environment variable $LD_LIBRARY_PATH is used, where
the library locations for EPICS, MXSlowControl and the modules have to be
specified. Additional environment variables must also be defined to determine
the module locations. In addition, the executables can be configured by defin-
ing the appropriate environment variables. For example, the list of destination
addresses for CA clients can be specified with the EPICS environment variable
EPICS_CA_ADDR_LIST. An example of thisEPICS.sh and the environment vari-
ables definition can be found in code 5.7.
5.5.3 Execution via System Control Manager
In most cases MXSlowControl runs permanently on the control machines. In addi-
tion, each machine must automatically resume its work after a restart. Therefore,
the system must handle the execution of MXSlowControl itself. For this purpose,
the Debian system manager systemd is used. This offers the possibility to initiate
processes by system start in parallel. For this the gerneric system management
operation systemctl is used [Deb]. To declare the start and end of the execution
of a command running in the background, a systemctl configuration file must
be created and loaded into the system. An example of such a file can be found in
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Chapter 5 - Development of the MAGIX Slow Control System
Code 5.7: Example of a thisEPICS.sh file
#EPICS
export EPICS_ROOT=/opt/epics
export EPICS_BASE=${EPICS_ROOT}/base
export EPICS_HOST_ARCH=$(${EPICS_BASE}/startup/EpicsHostArch)
export EPICS_BASE_BIN=${EPICS_BASE}/bin/${EPICS_HOST_ARCH}
export EPICS_BASE_LIB=${EPICS_BASE}/lib/${EPICS_HOST_ARCH}
if [ "" = "${LD_LIBRARY_PATH}" ]; then
export LD_LIBRARY_PATH=${EPICS_BASE_LIB}
else
export LD_LIBRARY_PATH=${EPICS_BASE_LIB}:${LD_LIBRARY_PATH}
fi
export PATH=${PATH}:${EPICS_BASE_BIN}
export HOST=$(hostname)
#Modules
export ASYN=${EPICS_ROOT}/modules/asyn
export STREAM=${EPICS_ROOT}/modules/stream
export AUTOSAVE=${EPICS_ROOT}/modules/autosave
export DEVGPIO=${EPICS_ROOT}/modules/devgpio
#PYEPICS
export PYEPICS_LIBCA=${EPICS_BASE_LIB}/libca.so
#MAGIX Modules
export FRONTEND=${EPICS_ROOT}/modules/frontend
#MXSlowControl
export MXSLOWCONTROL=${EPICS_ROOT}/MXSlowControl
export MXSLOWCONTROL_BIN=${MXSLOWCONTROL}/bin/${EPICS_HOST_ARCH}
export MXSLOWCONTROL_LIB=${MXSLOWCONTROL}/lib/${EPICS_HOST_ARCH}
export IOC=${MXSLOWCONTROL}/iocBoot/iocMXSlowControl
if [ "" = "${LD_LIBRARY_PATH}" ]; then
export LD_LIBRARY_PATH=${MXSLOWCONTROL_LIB}
else
export LD_LIBRARY_PATH=${MXSLOWCONTROL_LIB}:${LD_LIBRARY_PATH}
fi
export PATH=${PATH}:${MXSLOWCONTROL_BIN}
#Channel Access List
export EPICS_CA_ADDR_LIST="$(<${EPICS_ROOT}/CA_ADDR_LIST.local)"
#caArchiver
export CA_ARCHIVER_LOG=${EPICS_ROOT}/log
code 5.8. After loading the epics.service file, MXSlowControl can be started
with the command systemctl start epics. To initialize MXSlowControl at
every system start, the service has to be enabled with systemctl enable epics.
One problem is, that the IOC shell can not be accessed if MXSlowControl is a pure
background process. For several reasons like debugging, changing configurations
or some testings this access is required. To solve this the MXSlowControl IOC
Shell is running in a tmux session. The tmux program provides access to a number
of windows from the command line. It is structured as a server-client system.
When a tmux session is started, a server is automatically created to manage the
session. Any window opened in a session can be accessed from any number of
clients [Deb]. By combining tmux and systemctl, the IOC shell can be invoked
from the terminal after MXSlowControl has been initialized. Another advantage
is that this session can be accessed by multiple clients at the same time, so that
the information in the IOC shell can be retrieved from different locations.
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5.5 MAGIX Slow Control System
Code 5.8: Example of the epics.service file
[Unit]
Description=MAGIX EPICS IOC
After=network-online.target
After=epicsrepeater.service
[Service]
Type=forking
User=mxuser
Group=mxuser
TimeoutStartSec=5
ExecStart=/usr/bin/tmux new-session -ds EPICS -c
↪→ /opt/epics/MXSlowControl/iocBoot/iocMXSlowControl \
'source /etc/profile.d/epics.sh; \
$EPICS_ROOT/MXSlowControl/iocBoot/iocMXSlowControl/st.cmd;'
ExecStop=/usr/bin/tmux send-keys -t EPICS Enter 'exit' Enter
RemainAfterExit=yes
[Install]
WantedBy=multi-user.target
5.5.4 Defining UDEV Rules
The automatic detection and initialization of the connected hardware is managed
by the udev system. By default, new devices can only be accessed as super
user [Deb]. This means that to control devices via MXSlowControl, the IOC
must be run as super user. This is not recommended for several reasons such
as security. There are several solutions for this problem. The solution used for
MXSlowControl is to define udev rules to configure the hardware for its use at
initialization.
After the hardware is detected, basic information is read back from the devices,
called attributes. Based on the attributes udev configures the devices. Most
devices have unique attributes and it is possible to distinguish between them.
For example, to get the attributes for a particular device, the command udevadm
info -a -n /dev/$YOUR_DEVICE can be used. An example of the result of this
command can be found in code 5.9. To change the configuration for a specific
device, a new udev rule must be defined. You can define, for example, the user
or group that gets access to the device. Also, unique system links can be created
to identify and load the devices by their individual names into the software. An
example of such a rule can be found in code 5.10. Those rules must be defined
in a file with the extension .rules in the folder /etc/udev/rules.d/. For each
device, all rules defined in this directory are applied by udev to a device. When
defining rules, it is important to note that rules can overwrite each other. For
MAGIX, we have therefore created a global file that contains the rules for all
devices. This file is also installed via the installation script (see Section 5.5.1
and code C.1). It is therefore important that the rules of new devices are added
to this global file.
5.5.5 Implementation of the Bash/ZSH Completion
For the standard user it is sufficient to control EPICS via a GUI. However, ex-
perienced users and developers very often use the command line commands to
access PVs. Especially, in driver development when there is no GUI yet, develop-
ers depend on it. As the names of PVs can be long and complex strings, entering
them is often a pain. EPICS itself does not have a bash completion. In the con-
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Chapter 5 - Development of the MAGIX Slow Control System
Code 5.9: Example using the udevadmin command to read out the attributes
of a connected device
KERNELS=="ttyUSB0"
SUBSYSTEM=="tty"
ATTRS{interface}=="USB-RS485 Cable"
ATTRS{idProduct}=="6001"
ATTRS{idVendor}=="0403"
ATTRS{serial}=="FT03G5K6"
...
Code 5.10: Example for a udev rule.
SUBSYSTEM=="tty", ATTRS{idVendor}=="0403",
↪→ ATTRS{idProduct}=="6001", ATTRS{serial}=="FT03G5K6",
↪→ GROUP="mxuser", SYMLINK+="MPT200"
text of this thesis a bash completion was developed for the standard commands
like caget. Here, in addition to bash, the completion was also developed for the
shell ZSH, as most EPICS developers in the MAGIX group prefer it. However,
since the principle is identical and only the language differs, we will not make a
distinction between the two completions in this section.
The current version of the bash completion needs some preparation by the IOC.
The IOC lists all loaded records using the IOC shell command dbl. This list
can be stored in a file. Each startup script of the IOC uses this to create a file
named ${EPICS_ROOT}/dbl.txt including the list of all records provided by this
IOC. The bash completions uses this file as input to fill the array COMPREPLY. The
completion system retrieves this array to perform the completions or to output
the options. This allows the user to complete the PVs of the locally running
IOC. The completion also uses a second file, ${EPICS_ROOT}/dbl_global.txt,
as input. All PVs defined in this file can also be completed. The content of this
global file is up to the user. In most cases it will contain the db list of the other
IOCs in the subnet. These can be copied from the hosts using the scp command,
for example, and added to the dbl_global.txt file. The scripts for the Bash and
ZSH completion can be found in code C.4 and C.5
A new version for Bash completion is currently under development. Section 5.5.7
introduces the MXFrontend module. This allows to query the provided PVs of
each IOC over the network. The new version of Bash completion will make use of
this instead of querying the PV list from a file as before. How these two parts can
work together without causing a delay due to different query times is currently
being tested. Development will continue beyond the period of this work.
5.5.6 Integration of an Archiver (MAGIX executable)
EPICS does not have its own archiving program, but it does provide a powerful
tool, camonitor. This tool is connected to a list of PVs and prints the changes
of each PV in stdout with the timestamp, PV name and current value. If a
PV connection can not be opened or is lost, it attempts to reconnect to the PV
and prints a brief information. To archive this information, a simple solution
is to redirect the output to a file using the command camonitor $PV_LIST >
$LOG_FILE. This solution is very rudimentary and is not a definitive solution for
an archiver. However, it is very useful when you need to log some data quickly.
Since EPICS is an open source, we could create an archiving program for MAGIX
based on the camonitor program. The changes we made are described below.
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5.5 MAGIX Slow Control System
First of all, we changed the argument handling. Instead of expecting a list of
PVs, the caArchiver expects a file with a list of all PVs to be logged. For each
PV in the list, a connection is opened and an event callback function is attached.
The callback function handles the return for each change to a PV. This function is
modified to output the information to a file instead of the stdout. For each new
day, a new log file is opened with the corresponding timestamp. This prevents the
log files from becoming too large. Finally, we have added the ability to configure
the log directory via an environment variable called CA_ARCHIVER_LOG.
Despite the minor modifications, we managed to turn the camonitor tool into a
very stable and powerful archiver caArchiver. This is started via systemctl in a
tmux session like the IOC application. Only the file $CA_ARCHIVER_LOG/log.txt
must be created and filed with a list of PVs.
5.5.7 Development of the MXFrontend (MAGIX module)
The basic idea of the MXFrontend module is to read internal information of the
IOCs or to identify them via the network. For this purpose, the module extends
the IOCs with the appropriate functionalities. The communication principle is
based on that of Channel Access. MXFrontend opens an additional UDP socket
for each IOC, which waits for an external request. If a corresponding client makes
a request, the IOC opens a TCP connection to communicate with the client.
Although the module is still in development, it is already loaded and used by all
MXSlowControl IOCs. So far, the module supports only a few different requests.
One of them is the query of the status of an IOC (on or off). This function can
also be used to query which IOCs are currently running via an UDP broadcast
throughout the subnet. Another possibility is to poll the PV list of one or more
IOCs. Each IOC then sends this list to the client via the network, which can pro-
cess it. In addition, MXFrontend provides a command that queries all PVs in one
or more subnets and stores them in the file $EPICS_ROOT/dbl_global.txt. All
parameters like the UDP or TCP port numbers can be controlled by environment
variables. These are read by the system at startup and used for configuration.
The complete communication structure is already programmed and based on this,
other queries or commands can be sent to the IOC. Two further functions are
currently in preparation. The first one is the possibility to restart an IOC via the
network. The second function is to pass the information that the IOC currently
outputs as a stdout to a client. There are no limits for this module and so
one could also think about controlling certain IOCs completely via the module
MXFrontend. This means to send the configurations remotely to the IOC and
start it. Development of this module will also continue beyond the time period of
this work.
5.5.8 Expansion by the dbLoadMultiRecords Function
EPICS Base provides two functions for loading and configuring a db file. The
basic idea is to create db files with variables as placeholders. These variables
must be defined when loading a db file. The first IOC shell command to load
and configure a DB file is dbLoadRecords. This command takes two arguments,
first the db file to be loaded and second a pattern for the variable definitions.
An example of dbLoadRecords can be found in code 5.1. If a database is to
be loaded with different configurations, there are two possible solutions. First,
the dbLoadRecords command can be executed multiple times to load the same
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Chapter 5 - Development of the MAGIX Slow Control System
database file with different configurations. It should be noted that each loaded
record is created with a unique name as a new object in the database. If a name
is repeated, the second initialization is ignored [B3.14].
Additionally, the function dbLoadTemplate can be used. The command only
needs a template file as parameter in which all configurations are defined. The
configurations are stored as simple lists in the template file. The template function
can load multiple db files at once with a large number of different configurations.
An example of the dbLoadTemplate command can be found in code 5.11 [B3.14].
Code 5.11: Example of the dbLoadTemplate command. The file
devADC1248_mxrpi-14.template is loaded by the IOC shell. This contains
several configuration patterns for different db files. One of these definitions
is shown for the db file devADC1248Digital.db. The digital inputs for the
different channels are connected to the GPIO pins of a RPi. The mapping
between the channel number and the GPIO pin is defined in the template file.
So, each channel is connected to the corresponding GPIO pin after loading
and can be read out.
IOC shell command
dbLoadTemplate("devADC1248_mxrpi−14.template")
FILE: devADC1248_mxrpi−14.template
file devADC1248Digital.db {
pattern{P,Ch,GPIO}
{mxrpi−14,0,15}
{mxrpi−14,1,17}
{mxrpi−14,2,18}
{mxrpi−14,3,27}
{mxrpi−14,4,23}
{mxrpi−14,5,22}
}
...
FILE: devADC1248Digital.db
record( bi, "$(P):ADC1248:Input:Digital:Ch$(Ch)")
{
field(DTYP, "devgpio")
field(INP, "@GPIO$(GPIO)␣H")
field(SCAN, ".2␣second")
field(ZNAM, "low")
field(ONAM, "high")
}
...
Both options have a disadvantage. If a large number of records with one or more
consecutive numberings must be loaded, you have to write all possible configura-
tions into the IOC start script or template file manually. This may still be possible
for smaller sequences, but is a time-consuming and error-prone task for larger se-
quences. For example, 1024 identical records with sequential numbers had to be
loaded for a fiber detector. For this reason, another possibility for loading db files
was integrated in the context of this dissertation. This IOC shell command is
called dbLoadMultiRecords.
Like the dbLoadRecords command, the dbLoadMultiRecords command takes
two arguments: the db file to load and a pattern for the variable definitions. The
interpretation of the pattern has been changed to allow sequences. The pattern
is interpreted and for each sequence the db file is loaded with the corresponding
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5.5 MAGIX Slow Control System
pattern. For a sequence of numbers with the notation parameter1=1..8, the db
file is loaded with the patterns
parameter1=1
parameter1=2
...
parameter1=8.
To load the db file with a sequence of strings or non-continuous numbers, the
notation parameter2={val1;val2;val3} is used to generate the patterns:
parameter2=val1
parameter2=val2
parameter2=val3.
Multiple sequences can be used for one pattern. All possible combinations are used
to configure and load the db file. For example, the pattern P={device1;device2},
channel=1..8 is interpreted as:
P=device1, channel=1
P=device1, channel=2
...
P=device1, channel=8
P=device2, channel=1
P=device2, channel=2
...
P=device2, channel=8.
Two examples are given in code 5.12. The dbLoadMultiRecords command is
an useful extension to the standard db load commands and is used for many
MXSlowControl devices.
Code 5.12: Example of the command dbLoadMultiRecords. With the first
IOC shell command, dbLoadMultiRecords, the file devNewStepSB.db is
loaded three times with the three different axis numbers 1,3 and 5. With the
second IOC shell command the file devNewStepSB.db is loaded four times with
the different channel numbers 1 to 4.
IOC shell command
dbLoadMultiRecords("devNewStepSB.db","P=NewStep,␣cID=0,␣Axis={1;3;5},␣PORT=xyzTable")
dbLoadMultiRecords("devHMP4040.db","P=HMP4040,␣PORT=HMP4040,␣CH=1..4")
FILE: devNewStepSB.db
record(longout, "$(P):Axis$(Axis):SetPosition")
{
field(DESC, "Set␣Position␣of␣Axis$(Axis)␣Absolut")
field(DTYP, "stream")
field(OUT, "@devNewStepSB.proto␣setPos($(cID),$(Axis))␣$(PORT)")
field(LOPR,"−60000")
field(HOPR,"120000")
field(EGU, "micro−steps")
}
FILE: devHMP4040.db
record(ai, "$(P):$(CH):getVoltage") {
field(DTYP, "stream")
field(EGU, "V")
field(PREC, 3)
field(INP, "@devHMP4040.proto␣getDouble($(P),$(CH),MEAS:VOLT)␣$(PORT)")
field(SCAN, "1␣second")
field(PINI, "YES")
}
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Chapter 5 - Development of the MAGIX Slow Control System
5.5.9 Development for Profibus Support (StreamDevice BUS API)
Some MAGIX devices to control the gas jet target, like the pumping station or
the solenoid valve control can only be accessed using Process Field Bus (Profibus).
Since there is no EPICS module that allows communication with Profibus, we had
to develop a solution ourselves. Instead of writing our own device support, we
decided to write a StreamDevice BUS interface. For this purpose StreamDevice
provides an Application Programming Interface (API), especially a C++ interface
called StreamBusInterface. To add a new BUS, it is necessary to create a new
class as inheritance of StreamBusInterface. Then, the necessary methods have
to be implemented, such as connect, write or read. This is the place where the
communication via Profibus is specified. Finally, the new class must be registered
as a new StreamDevice bus. Then, the new bus can be used in EPICS [SDev2.8].
The advantage of this solution is that the Streamdevice environment can be used.
This means that the messages can be defined in the proto file and streamdevice
takes care of the communication via Profibus. An example for the usage of the
Profibus StreamDevice bus is given in code 5.13.
5.5.10 MXSlowControl Tutorial
For unexperienced EPICS developers, a tutorial was created in the internal group’s
mediawiki as part of this work. This tutorial is constantly updated to support
users in getting started with EPICS and MXSlowControl easier and to provide
additional professional tips for more complex tasks. The current version covers
the topics of process variables, StreamDevice, and PyEpics. Since almost all
devices can be controlled via StreamDevice, any user can integrate and use new
devices in MXSlowControl after completing this tutorial. Each section of the
tutorial covers the most important content and necessary information. Besides
examples, tasks are defined. Each solution of the defined tasks leads the user to
an important learning step. If a user gets stuck with a task, suggested solutions
can be displayed. Other topics that are still under construction are the creation
of OPIs with CSS, the creation of device supports, custom modules and more.
The goal is to create an all encompassing guide that can be used in the final
experiment to allow all experimenters to work with EPICS, quickly fix problems,
or add new devices.
5.5.11 Development of Device Support for MAGIX Built Devices
MAGIX has also built some self-developed devices for the experiment, which must
be controlled via EPICS. For these devices, you have to dive deep into driver de-
velopment, since most components only understand a very rudimentary language.
The electronics required for the application were developed in cooperation with
the in-house electronics workshop. To be able to read information from built-in
chips, the corresponding registers usually have to be set and read out. In this sec-
tion, we will look at two of these devices and get a basic overview of how device
support is developed for such a device.
ADC 1248
The device ADC1248 is a high frequency 24-Bits analog-to-digital converter (ADC)
developed from a MAGIX group member [Bir21]. In total eight differential ADC
inputs are provided. In addition, it provides six digital entries for signals of 3.3V
and one relay. All the electronics are placed on a board that can directly be
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5.5 MAGIX Slow Control System
Code 5.13: Example for the Profibus StreamDevice bus. The db is loaded
with two variables. The variable PROFIBUS contains the system link, the
destination address, the user data, the source address and the baud rate.
The baud rate is configured by an index. In this example the baud rate
baudrates[5]=1.5Mbps is set. The record calcout contains the information
to perform the communication. The DTYP field specifies that StreamDevice
is used for communication. The OUT field specifies Profibus as StreamDevice
BUS. The protocol set defines the structure of the message. Each valve cor-
responds to one bit. The message is a 4 byte long word to configure the entire
valve setting. For each bit that has the value 1, the corresponding valve is acti-
vated. All valves whose corresponding bit has the value 0 are then deactivated.
IOC shell command
# PROFIBUS=Device:Destination Address:User Data:Source Address:Baudrate
dbLoadRecords("db/devVUVG.db","P=VUVG,␣PROFIBUS=/dev/VUVG:3:840000:1:5")
FILE: devVUVG.db
record (bo, "$(P):Valve1")
{
field(ZNAM, "off")
field(ONAM, "on")
field(FLNK, "$(P):AllValves")
}
...
record(calcout, "$(P):AllValves")
{
field(DESC, "Set␣all␣Valves␣at␣once")
field(DTYP, "stream")
field(OUT, "@devVUVG.proto␣set␣Profibus␣$(PROFIBUS)")
field(INPA, "$(P):Valve1")
field(INPB, "$(P):Valve2")
field(INPC, "$(P):Valve3")
field(INPD, "$(P):Valve4")
field(INPE, "$(P):Valve5")
field(INPF, "$(P):Valve6")
field(INPG, "$(P):Valve7")
field(INPH, "$(P):Valve8")
field(INPI, "$(P):Valve9")
field(INPJ, "$(P):Valve10")
field(CALC, "A∗1+B∗2+C∗4+D∗8+E∗16+F∗32+G∗64+H∗128+I∗256+J∗512")
field(SCAN, "Passive")
}
FILE: devVUVG.proto
set {
out "%#0.4r";
in "%∗1r";
}
mounted on a RPi. For the communication the board is connected to the GPIO
header of the RPi. To develop the control via EPICS was part of this thesis.
The digital entries and the relay can be controlled using the dev-gpio module (see
Section 5.3.5), so we will focus below on the development of the device support
to control the ADC inputs.
Two ADS1248 chips [ADS] are used for the ADC. The registers of the chips are
controlled through the Serial Peripheral Interface (SPI). To read the voltage for a
channel, the chips must be configured. This is managed by the chip registers. Since
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Chapter 5 - Development of the MAGIX Slow Control System
the communication is too complex for StreamDevice, we decided to build a device
support for this part. For this purpose, an individual C++ class was developed
for each register to manage the corresponding information. An example of such
a class can be found in code C.6 and C.7. All register classes are combined into
one class representing the ads1248 chip. the ads1248 class also handles the SPI
communication for all registers. This allows the chip to be configured as desired
and the voltage can be read back from the chip. Up to this point the development
is independent of EPICS and the chip can be controlled by different programs
using this driver. To read out the voltage with EPICS, a device support has been
developed (see Section 5.2.1). An example of a data set for reading out the voltage
can be found in code 5.14. The complete device support is packaged in a module
that can be installed on an RPi if needed. After installation the ADC1248 board
can be controlled.
Code 5.14: Example of a record using the ADC1248 module. The getVoltage
function of the ADC1248 module takes as paramter the channel to be read, the
gain and the sampling rate. Since different voltage dividers can be placed on
the ADC the voltage needs to be calculated using this information as well. The
db file is loaded via template since each channel can be configured individually.
IOC shell command
dbLoadTemplate("$(TOP)/db/devADC1248_RPi0.template")
FILE: devADC1248_RPi0.template
file ../../db/devADC1248Analog.db {
pattern{P,Ch,Divider,Gain,SPS,OFFSET}
{RPi0,0,5,1,20, 0}
{RPi0,1,5,1,20, 0}
{RPi0,2,5,1,20, 0}
{RPi0,3,5,1,20, 0}
{RPi0,4,5,1,20, 0}
{RPi0,5,5,1,20, 0}
{RPi0,6,5,1,20, 0}
{RPi0,7,5,1,20, 0}
}
FILE: devADC1248Analog.db
record(longin, "$(P):ADC1248:Input:Analog:Ch$(Ch):Raw")
{
field(DESC,"Raw␣value␣of␣the␣voltage␣for␣Channel␣$(Ch)")
field(DTYP,"ADC1248")
field(SCAN,"1␣second")
field(INP,"getVoltage␣$(Ch)␣$(Gain)␣$(SPS)")
field(FLNK,"$(P):ADC1248:Input:Analog:Ch$(Ch):Voltage")
}
record(calc, "$(P):ADC1248:Input:Analog:Ch$(Ch):Voltage")
{
field{DESC,"Voltage␣of␣Channel␣$(Ch)␣with␣voltage␣divider␣1/$(Divider)"}
field(INPA,"$(P):ADC1248:Input:Analog:Ch$(Ch):Raw")
field(CALC,"$(Divider)∗2.048␣/␣(␣1<<23␣)∗A␣−␣$(OFFSET)")
field(SCAN,"Passive")
}
MXSiDet
Another large device carrier module is the MXSiDet module. To control a sil-
icon strip detector (see Section 4.2.3), a front-end board was developed by the
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5.6 Slow Control Development for the Silicon-Strip Detector Test Stand
electronics workshop. On the board are several chips, which control e.g. the trig-
ger threshold or the temperature. The whole communication is based on Inter-
Integrated Circuit (I2C). Several front-end boards (silicon strip detectors) can be
controlled by an I2C controller via an I2C-Multiplexer (I2C-MUX). The develop-
ment of the entire slow control driver for this front-end board was part of this
work. As in the case of the ADC1248, the entire communication is too complex
for a StreamDevice. Therefore, a separate device support module was developed
to control the whole device. A detailed description of this driver can be found in
the Section 5.6.3. This is part of the next section which explains the design and
development of the entire slow control for the silicon strip detector test bench.
5.6 Slow Control Development for the Silicon-Strip De-
tector Test Stand
As mentioned in Section 4.2.3 currently one silicon strip detector is being tested
by the MAGIX group (see Section 4.2.3). The construction of a spatially resolving
silicon strip detector system was the topic of a master thesis in the MAGIX group
[Gei20]. In this thesis, the initial development of the detectors electronics and the
construction of a test stand are described as well as a discussion of the results of the
first measurements. In this section, we will focus on the slow control development
for this test bench, which is a part of this thesis.
The test stand consists of a vacuum chamber equipped with several supply lines to
enable operation of the detector. To achieve a vacuum of the order of 10−5 mbar,
a rotary vane pump and a turbomolecular pump are connected to the chamber.
The detector assembly itself is mounted inside the chamber and consists of the
X1 chip directly connected to the manufactured readout electronics and the APV.
An MPT200 device from Pfeiffer Vacuum is used to control the chamber pressure.
The low voltage supply to the detector and the vacuum transmitter is controlled
by the HMP4040 device from Rohde & Schwarz.
This section is used as a example of how to integrate different types of devices
into EPICS. First, we will see how to read a MPT200 connected via the serial
port. Then, we will discuss how the network device HMP4040 can be controlled
via EPICS. Finally, the frontend board of the silicon strip detector (SiDet) will
be explained in more detail as well as the development of a corresponding device
driver. The general structure from the point of view of the control software is
shown in Fig. 5.5.
5.6.1 Readout of the MPT200 Device via StreamDevice ttyUSB
The MPT200 device is a pressure sensor that can be controlled via RS-485. To
connect the MPT200 to a RPi a RS-485 to USB converter cable is used. For the
communication with the RS-484 so called telegram frames are used. They consist
only of ASCII code characters [MPT200]. This makes this device a fantastic
candidate to be integrated into EPICS using the StreamDevice module.
First, the asynPort driver needs to be configured. Since the device is connected
via serial port the IOC shell command drvAsynSerialPortConfigure is used for
this purpose. The configuration for the serial port can be taken from the device
documentation as [MPT200]:
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Chapter 5 - Development of the MAGIX Slow Control System
CSS
(GUI)
Channel Access
Network
RaspberryPi HMP4040
IOC USB Power Supply
GPIO Header
±5V 24V
I2C
RS-485
Readout Electronics MPT200
Figure 5.5: The test stand of the silicon strip detector is controlled by an IOC
running on a RPi. The low voltage power supply HMP4040 is controlled via
network. The pressure detector MPT200 for the test chamber is controlled via
RS-484 and directly connected to the RPi via USB. The readout electronics is
controlled via I2C. The support voltage is provided by the HMP4040.
• 9600 baud
• 8 data bits
• 1 stop bit
• no parity.
An example using asyn to specify this configuration is already given in code 5.4
using the udev rule code 5.10.
Telegram frames are used for the communication. The structure of this commu-
nication frame is [MPT200]
a2 a1 a0 * 0 n2 n1 n0 l1 l0 dn . . . d0 c2 c1 c0 cR
with
Data Byte Description
a2 - a0 Address of MPT200
* Action (0 for data request; 1 for control command or data
response)
n2 - n0 Control Command
l1-l0 Data length of dn . . . d0 as integer
dn - d0 Data in the corresponding data type
c2 - c0 checksum (sum of ASCII values of a2 to d0) modulo 256
cR Terminator: Carriage return (ASCII 13).
The address of each MPT200 can be set directly with a switch at the device. The
control parameters and the data types are given in the device documentation.
This frame must be translated into a StreamDevice protocol. An example for
the command read pressure P:740 is already given in code 5.5. The checksum is
calculated using the StreamDevice converter %+<sum8>. Hereby, sum8 defines that
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5.6 Slow Control Development for the Silicon-Strip Detector Test Stand
the sum of all characters are calculated modulo 28. The character + is defined to
represent the checksum as decimal ASCII [SDev2.8].
The pressure is returned as U_Expo_New type with a size of 6 characters. The
U_Expo_New integer number nnnnee can be converted to a float in units of mbar
using the formula [MPT200]
p = nnnn ∗ 10ee−23 [mbar].
To convert the pressure in mbar a calc record is used (see code 5.15).
Code 5.15: Example of a calc record converting U_Expo_New format to mbar
record(calc, "$(P)$(R)Pressure")
{
field(DESC, "Calculate␣Pressure␣of␣device␣MPT200␣in␣mbar")
field(INPA, "$(P)$(R)GetPressureRaw")
field(CALC, "FLOOR(A/100)∗10^(A%100−23)")
field(SCAN, "Passive")
field(EGU, "mbar")
field(PREC, 7)
}
The further control commands can be translated into a protocol and linked to a
record in the same way. The conversions are then also done via calc records.
5.6.2 Controlling the HMP4040 Device via StreamDevice TCP
The low voltage supply for the entire test stand is provided by the HMP4040
network device. Each of the four output channels can easily be controlled from
the front panel. There is also a small display on the front panel that shows some
information. Although front panel control may be sufficient for a test stand, the
device is connected to the network via the built-in Ethernet port to allow remote
control. After configuring the HMP4040 TCP/IP network protocol, the unit can
be controlled over the network. All control commands consist of ASCII code
characters [HMP]. Thus, the HMP4040 is also an excellent device that can be
controlled by EPICS with StreamDevice.
First the asynPort driver must be configured. Since the HMP4040 is a network
device, the IOC shell command drvAsynIPPortConfigure is used for this. To
configure asyn, the required information can be obtained from the device docu-
mentation or set on the front panel of the device [HMP]. With the information
• Network Protocol: TCP
• IP address: 10.32.108.213
• Port Number: 5025 (default)
the asynPort can be configured as:
drvAsynIPPortConfigure("HMP4040","10.32.108.213:5025␣TCP",0,1,0)
The structure for the control commands of the HMP4040 device is simple. First,
the output channel must be selected, then the control command can be sent.
For example, to set the output voltage for channel 1 to 25V, the following two
commands must be sent to the device [HMP]
INST OUT1
VOLT 25.0
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Chapter 5 - Development of the MAGIX Slow Control System
To read a value, the control command is extended by the character ?. For example,
to read the current voltage for channel 2, the following two commands must be
sent [HMP]
INST OUT2
MEAS:CURR?
The response is only the value of the current in ASCII characters, e.g. 1,200, in
units of A.
Several commands can be sent to the device at once. For this purpose the line
feed character \n (ASCII 10) must be used as terminating character between the
commands. The structure of the individual commands for this device is identical.
They differ only in the corresponding data types that must be transmitted. Since
a large number of commands must be integrated into EPICS, it makes sense to
create a separate StreamDevice protocol for each of the different data types and
to pass the commands as an argument through the records. Apart from reading
the identifier as a string, the HMP4040 only uses the double, integer and boolean
data types for transmission. This results in only seven different protocols having
to be written for the HMP4040: getID, getDouble, getInt, getBool, setDouble,
setInt, and setBool. An example of the records and protocols for setting the
voltage and reading back the current can be found in code 5.16. In addition to
the command, the channel number is also an argument to the protocols. Since the
channel number is also a parameter of the database, each record has to be defined
only once. With the IOC shell command dbLoadMultiRecords (see Section 5.5.8)
all four channels will be loaded into the IOC.
5.6.3 Driver Development to Control the SiDet board via RPi I2C
This section explains how the driver for the readout electronics of the silicon
strip detector array (see Section 4.2.3) was built. Figure 5.6 shows the readout
electronics developed in cooperation with the electronics workshop for the SiDet.
Since a space-saving design is required for all parts in the scattering chamber,
the electronics are divided into two parts, the frontend-board and the HDMI
distribution board. The frontend-board is mounted directly under the detector in
the vacuum chamber (see Fig. 4.13). All the electronics to control the detector
are located there. An overview of the integrated circuits (ICs) placed on this
frontend-board is given in Table 5.1. The HDMI distribution board is located
outside of the vacuum chamber. It provides the supply voltage of ±5 V and an
I2C interface to the frontend-board via an HDMI cable. Additionally, an external
trigger signal can be input or the generated trigger signal and the analog signals of
the frontend-board can be output via LEMO sockets. To control the electronics,
the I2C master is connected to the HDMI contribution board [Gei20].
A RasperryPi is used as I2C master. This can be connected directly to the HDMI
board via the GPIO header. To control multiple boards with one master an I2C
multiplexer based on the IC PCA9548AD can be used. This allows to control up
to 8 silicon strip detectors with one I2C master [Gei20].
The SiDet library
The first step in creating a device driver is to create a C++ library that enables the
communication with the device. The advantage of an EPICS independent library
is, that the device can be controled with different programs. This allows flexible
use of the device and simplifies the distribution of the driver to the community.
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5.6 Slow Control Development for the Silicon-Strip Detector Test Stand
Code 5.16: Example of the db and proto file for the HMP4040 device. The db
is loaded with dbLoadMultiRecords to initialize the four channels separately.
Each record is associated with the protocol for the corresponding data type.
The channel number and command are passed as arguments to the protocol.
Each protocol selects the output channel and sets or requests the values for the
passed commands. Since the set values can be read back, each setting protocol
requests the current value from the HMP4040 device during initialization using
the @init macro.
IOC shell command
dbLoadMultiRecords("devHMP4040.db","P=Silicon:LV,␣PORT=HMP4040,␣CH=1..4")
FIILE: devHMP4040.db
record(ai, "$(P):$(CH):getCurrent") {
field(DESC, "read␣the␣current␣value␣of␣channel␣$(CH)")
field(DTYP, "stream")
field(EGU, "A")
field(PREC, 3)
field(INP, "@devHMP4040.proto␣getDouble($(P),$(CH),MEAS:CURR)␣$(PORT)")
field(SCAN, "1␣second")
field(PINI, "YES")
}
record(ao, "$(P):$(CH):setVoltage") {
field(DESC, "set␣voltage␣value␣of␣channel␣$(CH)")
field(DTYP, "stream")
field(EGU, "V")
field(PREC, 3)
field(OUT, "@devHMP4040.proto␣setDouble($(P),$(CH),VOLT)␣$(PORT)")
}
...
FIILE: devHMP4040.proto
getDouble{ # $1=$(P); $2=${Channel}; $3=${Command}
out "INST␣OUT\$2\n\$3?\n";
in "%f";
}
setDouble { # $1=$(P); $2=${Channel}; $3=${Command}
out "INST␣OUT\$2\n\$3␣%f\n";
in "%∗f";
@init {
out "INST␣OUT\$2\n\$3?\n";
in "%f";
}
}
...
Since all chips can be accessed with I2C one class was built to handle the entire
communication via the I2C bus. To avoid problems caused by parallel read and
write accesses, a singleton class was written for this purpose. The special feature
of a singleton is that only one instance (one object) of the class can exist at the
same time. The singleton class provides a static method to return the instance of
this class. This allows other methods to access this object/class. The class for the
I2C communication is called RPiI2C. It provides methods to open or close the I2C
bus and to write or read data to/from the device. The source code of the class
RPiI2C can be found in code C.8 and C.9.
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Chapter 5 - Development of the MAGIX Slow Control System
Table 5.1: Overview of the integrated circuits (ICs) placed on the frontend-board of the
readout electronics for the silicon strip detector [Gei20].
IC Type Description
AD5693RBR 16-bit DAC Set threshold for the trigger signal [AD]
ADS122C04IP 24-bit ADC Measure the voltage drop from two PT1000 temperature
sensors [ADS2]
- 2xPT1000 Platinum Resistance Temperature Detectors;
Connected to the cooling plate of the frontend-board
and the detector
MCP9808T Digital Temper- Measurement of the local temperature on the frontend-
ature Sensor board [MCP]
Frontend-Board
HDMI-Board
±5V
I2C-Multiplexer Analog Trigger
out out/in
RaspberryPi
Figure 5.6: Overview of the SiDet readout electronics [Gei20]. A detailed
description can be found in the text.
Similar to the ADC1248 (compare Section 5.5.11), a separate class has been writ-
ten for each chip on the frontend-board to provide the complete communication.
Each of these classes uses the RPiI2C class to write or read data. The structure
of the classes is similar and differs only in the way the registers of the chips are
set or read and which data can be supplied. To describe the frontend-board, a
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5.6 Slow Control Development for the Silicon-Strip Detector Test Stand
class SiDet was created, which contains an object of the classes for each chip.
The requests to read or write the information are then automatically assigned to
the chips respectively the corresponding class objects and processed. In order to
include a possibly integrated I2C multiplexer, a further class for controlling was
added. This can be loaded if necessary also into the SiDet class. By creating
an instance of the SiDet class the complete frontend-board can be controlled. A
class diagram of the entire device driver can be found in Fig. 5.7.
In addition to the basic functions for controlling the device, another useful feature
has been added, namely a logging system. This was implemented by a class mxlog
based on the header-only library spdlog which was developed to control the entire
logging. Each class of the device driver has an individually configured instance
SiDet . . . 1 AD5693 . . .
private: private:
mxlog m_logger mxlog m_logger
int m_ChannelNumber uint8_t m_AD5693_I2C_ADDR
PCA9548A∗ m_MUX
AD5693 m_DAC 1 public:
ADS122C04 m_ADC setDAC(...)
MCP9808 m_PCB getDAC(...)
set_level(...)
public:
getPCBTemperature(...)
setThreshold(...)
getThreshold(...)
getResistance(...)
set_level(...) 1
ADS122C04 . . .
private:
. . . mxlog m_loggerRPiI2C uint8_t m_ADS122C04_I2C_ADDR
Singleton
public:
private: getValue(...)
static RPiI2C∗ m_device set_level(...)
// private constructor for Singleton
RPiI2C()
mxlog m_logger
int m_fd_I2C
public: 1
// return Instance of RPiI2C MCP9808 . . .
static RPiI2C& device()
open(...) private:
close(...) mxlog m_logger
readI2C(...) uint8_t m_MCP9808_I2C_ADDR
writeI2C(...) public:
set_level(...)
getTemp(...)
set_level(...)
mxlog . . .
private:
level_enum m_level 0..1
public: PCA9548A . . .
log(...) private:
trace(...)
debug(...) mxlog m_logger
info(...) uint8_t m_MUX_I2C_ADDR
warn(...) uint8_t m_channel
error(...) public:
critical(...) setMUXChannel(...)
set_level(...) getMUXChannel(...)
get_level(...) set_level(...)
Figure 5.7: Class diagram of the device driver for the silicon strip detector
array. Each detector can be controlled using the SiDet class. For each chip
an instance of the coresponding class is loaded. All communication is handled
by the singleton class RPiI2C. For logging the class mxlog is used.
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Chapter 5 - Development of the MAGIX Slow Control System
of mxlog. For each instance, a log level can be set that defines what information
should be output. So, it is possible to output only error messages or more detailed
messages up to trace level. Since each object can be configured separately, it
is possible to set the logging specifically for different application areas, which
e.g. greatly simplifies troubleshooting in case of problems. The log messages are
displayed in the terminal with different color codes. Each message contains a time
stamp, an entry about the class that issues the message, and the protocol level.
By this the log has a search functionality. An example of the output is given in
Fig. 5.8. The source code of the class mxlog can be found in code C.10 and C.11.
(a) Output with log level set to error for all logger.
(b) Output with log level set to debug for all logger.
(c) Output with log level set to trace for logger I2C.
Figure 5.8: Example output of the mxlog class for the silicon strip detector
device driver. The first column shows the timestamp. The second column
represents the class printing the message and the third one the message level.
Finally, the message is attached.
When creating a library, it is a good practice to write unit tests and create a
documentation. For each class, a unit test was created based on the header-
only library Catch2. Each unit test verifies the functionality and gives an error
message if something goes wrong. For the documentation the doxygen system was
used. This creates a documentation from the comments, which were entered in
the header files of the individual classes in a special syntax. This not only makes
it easier for users to access, but also helps developers find their way around the
code.
How to Build EPICS Modules and Link them to the Library
After installing the SiDet library, the module can be created to integrate the driver
into EPICS. With the EPICS basic command makeBaseApp.pl -t support a
directory structure for a new module can be created. Then the device support
(see Section 5.2.1) must be programmed. As shown in the example code 5.3
EPICS provides an interface where for an input record only the two functions
init_record and read_ai have to be designed. For an output data set, write_ai
must be defined instead of the read function. In the initialization function, a new
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5.6 Slow Control Development for the Silicon-Strip Detector Test Stand
instance of the SiDet class is created, if it does not already exist. The read or
write functions have access to the information stored in the INP or OUT field.
This can be used to specify the information that is controlled by the record.
In the case of the silicon strip detector, simple commands are defined, such as
getThreshold. If this command is defined in the INP field, the corresponding
record will receive the threshold value as a result. Similar commands are defined
for the other parameters. The full definition of these functions can be found in
code C.12. To connect the SiDet module to EPICS, a database definition must
be set. In it, the data type must be linked to a record. In addition, the device
support structure associated with the data type must be defined. The two entries
for the MXSiDet module are
device(ai,CONSTANT,devSIDET_ai,"SIDET")
device(ao,CONSTANT,devSIDET_ao,"SIDET")
Thus, the data sets can access the silicon strip detector data by configuring the
codewordDTYP as codewordSIDET. An example of reading the PCB temperature
and setting or reading the threshold is given in code 5.17.
Code 5.17: Example of some of the SiDet records. The DTYP field specifies that
the SiDet library should be used. The value in the INP and OUT fields specifies
what information is associated with the record. For example, getTemperature
specifies that the temperature of the PCB is to be read from the frontend-board
and stored in the VAL field of the record.
record(ai, "$(P):getPCBTemperature")
{
field(DTYP,"SIDET")
field(SCAN, "10␣second")
field(INP,"getTemperature")
field(PREC, 3)
field(EGU,"◦C")
}
record(ai, "$(P):getThreshold")
{
field(DTYP,"SIDET")
field(SCAN, "5␣second")
field(INP,"getThreshold")
field(PREC, 3)
field(EGU,"V")
}
record(ao, "$(P):setThreshold")
{
field(DTYP,"SIDET")
field(OUT,"setThreshold")
field(EGU,"V")
field(PREC, 3)
field(DRVL, 0)
field(DRVH, 2.5)
}
In addition, a new IOC shell command called MXSiDet_SetLogLevel has been
written to set the logging level of the mxlog. As the device support, EPICS
provides an interface for this purpose as well. After the appropriate C++ structures
have been defined, they must be registered in EPICS. This is also done via a dbd
entry with
registrar(SetMXSiDetLogLevelRegistrar)
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Chapter 5 - Development of the MAGIX Slow Control System
Now the new IOC shell function can be used. To set the log level for all devices,
the command
MXSiDet_SetLogLevel("log_level")
can be used. The strings trace, debug, info, warning, error, critical or off can be
used as log_level. With the exception of off, each log level specifies the minimum
level for the output. For example, if the level is set to info, warnings or errors will
also be displayed. To set the logging level for only one class, a second parameter
can be set
MXSiDet_SetLogLevel("log_level","device_name")
The strings SiDet, MUX, DAC, ADC, PCB or I2C can be used as device name.
The full definition of this IOC shell command is shown in code C.13.
Summary and Outlook
After the SiDet library is connected to EPICS via a module, the silicon strip
detector test stand can be fully controlled. The structure of the slow control was
created in such a way that further devices, which are later required for MAGIX,
can easily be integrated. In order to be able to control the test stand in the lab,
a corresponding GUI was created in CSS. An example of one OPI can be seen in
Fig. 5.10.
Figure 5.10: CSS OPI for the silicon strip detector test stand. The
Board Control display contains all the information associated with the SiDet
frontend-board. The pressure in the chamber, the voltage and the current of
the power supply are displayed in text fields. In addition to the PCB temper-
ature, the temperature of the detector and the cooling plate for the frontend-
board are displayed. For the two PT1000s, the resistances are also displayed.
The threshold value of the DAC can be read or set with the OPI. To have an
overview of the trend for all temperatures, a corresponding diagram is dis-
played on the bottom of the OPI.
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CHAPTER 6
Simulation of the
Electro-Disintegration Reaction
16O(e, e′α)12C
In this chapter, we will discuss the simulation of the cross section of the electro-
disintegration reaction 16O(e, e′α)12C. The objective of the simulation performed
as part of this thesis was to estimate the expected statistical errors of the electro-
disintegration reaction of 16O for the experimental setup in Phase 1 and 2 of the
planned measurements at MAGIX. Since no definite information on the efficiency
of the detectors and the beam parameters of MESA is available at this time,
background simulations could not be included in the simulation. In addition, the
acceptances of the detectors were determined purely geometrically because of the
efficiencies that still need to be determined.
In the first section, we will get a general overview of physics simulations. For this
purpose, we will take a closer look at the Monte Carlo integration, a powerful tool
used for simulations. In addition we will discuss the general idea of simulating
the cross section of a scattering experiment. Then, in Section 6.2 the simulation
written within this thesis for the reaction 16O(e, e′α)12C will be explained. All
necessary parts of the simulation as well as the analysis will be discussed. Finally,
the results will be presented and discussed.
6.1 Overview of Physics Simulations
Simulations have become powerful tools that can be used in a variety of ways. To
list the multitude of possibilities is beyond the scope of this thesis. In general, one
can say that simulations use the computing power of machines or even entire server
systems to perform calculations based on models. The results of these simulations
can or must then be processed and interpreted by the users. To understand
the evolution of galaxies, for example one can simulate different scenarios using
suitable models and by varying the input parameters. The results can then be
compared with astrophysical observations. Another possible application is the
simulation of an experiment. For example, if you want to estimate the expected
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Chapter 6 - Simulation of the Electro-Disintegration Reaction 16O(e, e′α)12C
count rate in a detector for a scattering measurement, you can simulate this
experiment. For this purpose, the cross section must be calculated for a given
detector acceptance. To achieve this, you usually have to integrate the differential
cross section. The desired count rates can then be determined from the results
of the simulation. A very useful tool to perform such an integration within a
simulation is the Monte Carlo integration.
6.1.1 Principle of Monte Carlo Integration
The Monte Carlo (MC) integration is a useful tool to calculate the value of a
multidimensional integral. In the following, we will see that the error of the MC
integration scales like 1 √/ N, where N is the number of evaluations, regardless of the
dimension. Even though the convergence of 1 √/ N is quite slow, MC integration is
one of the preferred methods for evaluating integrals with high dimensions [Wei00].
For a function f : Rn → R, with n ∈ N, the mean value in a finite volume V ⊂ Rn
is defined as [Mer21]
〈 〉 := 1
∫
f vol( ) f(x) d
nx .
V V
The mean value can be approximated using the formula [Mer21; Wei00]
∑N
〈f〉 ≈ 1 f(xi),
N i=1
with uniformly distributed xi ∈ V ∀i. So, the integral can be approximated with
[Mer21; Wei00] ∫ V ∑N
I := f(x) dnx ≈ f(xi),
V N i=1
with V ≡ vol(V ). This is called the Monte Carlo integral. The law of large
numbers ensures that the MC integral converges to the true value of the integral
[Wei00]
V ∑N
I = lim f(xi).
N→∞ N i=1
The error of the MC integration can be described using the variance σ2(f) of the
function f [Mer21; Wei00] ∫
σ2(f) : = 〈 (f〉(x)− I)2 dnxV
= f 2 − 〈f〉2 .
Therefore, we get [Mer21(; Wei00]:∫ )2V ∑N 2
f(xi)− dn =
V
I x σ2(f).
V N i=1 N
This means that the average error of the MC integration is given as
√V σ(f).
N
As result the MC∫integration can be written as∑N √
f(x) dnx = V f(xi)± √
V 〈f 2〉 − 〈f〉2. (6.1)
V N i=1 N
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6.1 Overview of Physics Simulations
This shows that the error of the MC integration scales like 1 √/ N and is independent
of the dimension n. The exact value of the variance cannot be determined, but
can be approximated using [Wei00]
2( ) ≈ 1
∑N
σ f (f(x 2 2i)) − 〈f〉 . (6.2)
N i=1
In practice the points xi are calculated using random numbers. Since true random
numbers do not exist on a computer the so called pseudo-random numbers are
used. Pseudo-random numbers are not generated randomly, but are generated
deterministically in the computer by a certain algorithm. Any sequence of pseudo-
random numbers should appear randomly to someone who does not know the
algorithm. Pseudo-random number generators can establish correlations between
sequentially generated numbers. Balling points can be generated in this way. This
problem must be considered when selecting a pseudo-random number generator.
To avoid errors caused by the generator, it is useful to perform the same MC
integration with different pseudo-random number generators [Wei00].
Furthermore, quasi-random numbers can be used. Quasi-random numbers are not
random numbers, but are generated by a numerical algorithm and designed to be
distributed as uniformly as possible to reduce the errors in the MC integration.
Even though quasi-random numbers are designed to sample an n-dimensional
space as uniformly as possible, they are not generally preferable to pseudo-random
numbers. Only if the integrand is sufficiently smooth it makes sense to use the
quasi-random numbers [Wei00]. For more information on pseudo-random and
quasi-random numbers please refer also to [Knu97; Nie92].
Now, we will perform a simple example. Therefore, the area of a circle with radius
r = 1 will be calculated using a MC integration. The area is given as
A = πr2 = π = 3.1415926 . . .
Using the function 
( ) = 1 for x2 + y2 ≤ 1f x, y 0 otherwise,
the area can also be calculated a∫s 1 ∫ 1
A = f(x, y) dx dy .
−1 −1
Using a MC integration the area can be approximated as
≈ 4
∑N
A f(xi, yi),
N i=1
with the random numbers xi, yi ∈ [−1, 1] and V = 2 · 2 = 4. The integration
was performed with different numbers of points N . Two different random num-
ber generators were used, one pseudo-random generator and one quasi-random
generator using the Sobol sequence. Figure 6.1 illustrates the distribution of the
random numbers for both generators. The results of the MC integration are given
in Table 6.1. In this table also the average error is given, approximated using
Eq. (6.2) to calculate the variance.
As we can see in this example, the error decreases as the number of generated
points increases. In general, one way to improve the MC integration result is to
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Chapter 6 - Simulation of the Electro-Disintegration Reaction 16O(e, e′α)12C
1 1
0.5 0.5
0 0
−0.5 −0.5
−1 −1
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1
(a) Pseudo-random: 1000 points (b) Pseudo-random: 10000 points
1 1
0.5 0.5
0 0
−0.5 −0.5
−1 −1
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1
(c) Sobol sequence: 1000 points (d) Sobol sequence: 10000 points
Figure 6.1: Illustration of the distribution of the random numbers using a
MC integration to calculate π. Two different generators were used, a pseudo-
random generator and a quasi-random generator using the Sobol sequence.
This example shows how the sobol sequence is used to uniformly sample the
space.
increase N as much as possible. However, since the error converges like 1 √/ N, one
quickly gets into orders of magnitude where computations take much more time
without making any significant contribution to the improvement.
On the other hand, the variance can be reduced using different techniques. One
of these techniques is the importance sampling. For simplicity, we assume n = 1.
The idea is to redu∫ce the varian∫ce by changing th∫e integration varaiables [Wei00]:
( ) d = f(x) f(x)f d x ( ) p(x) dx = ( )dP (x),p x p x
with ( ) = dP (x)p x d . The mean value of the integral is then calculated as [Wei00]x ∑N
E := 〈 〉 ≈ 1 f(xi)f ,
N i=0 p(xi)
with xi = P−1(yi) and yi ∈ V ′ = P (V ) the sampling point of the MC integration.
Then, the error of the MC integration be(com)es [Wei00]′
√V fσ .
N p
106
6.1 Overview of Physics Simulations
Table 6.1: This table shows the results of the calculation of π using a MC integration.
The first column indicates the number of grid points used for the calculation. The error is
calculated using Eq. (6.2). The difference ∆rel represents the deviation in percent. Two
random number generators were used, a pseudo-random generator and a quasi-random
generator based on the Sobol sequence. In the case of the pseudo-random generator, it
can be seen that the result for 100 sample points is closer to the true value than for 1000
or 10 000 points. Since the range is covered randomly, such effects may well occur. In
the case of the Sobol sequence, the results improve continuously as the number of events
increases. This is to be expected in this case, since the range is covered as uniformly as
possible. In both cases, the results improve as expected for large numbers.
Pseudo-Random Sobol Sequence
N πcalc errcalc ∆rel[%] πcalc errcalc ∆rel[%]
100 3.160 00 0.245 24 0.585 92 3.000 00 0.264 58 4.507 03
1 000 3.192 00 0.076 23 1.604 52 3.112 00 0.079 47 0.941 96
10 000 3.166 00 0.024 45 0.776 91 3.137 20 0.024 82 0.139 82
100 000 3.136 72 0.004 65 0.155 10 3.141 04 0.004 62 0.017 59
1 000 000 3.142 34 0.001 71 0.023 66 3.141 67 0.001 71 0.002 40
10 000 000 3.141 63 0.000 28 0.001 25 3.141 59 0.000 59 0.000 19
The relevant quantity for the error is thus the function f(x)/p(x). The variance can
be greatly reduced if p(x) is chosen to be as close as possible in form to f(x). The
variance can be approxima(ted)as [Wei00]( )2
2 f ∑Nσ ≈ 1 f(xi) 2
p N i=0 p(xi)
− E .
The function p(x) must be chosen carefully because it can also increase the vari-
ance and thus the error. For example, if the function p(x) is zero at one point x̂
and f(x) is not, then the variance becomes large for each xi ≈ x̂.
As an example of importance sampling, we now consider the integral of the func-
tion f(x) = 12 on the interval [ε, 10] w∫ ith a fix ε ∈ (0, 1] small:x 10
= 1I
ε x2
dx .
The function f(x) converges to zero for large x. For x→ 0 it diverges to +∞. The
dominating part of this integral is the integral on the region [ε, 1]. Analytically
the solution to this integral∫is10
= 1
[ ]10
d = −1 = 1 − 1I 2 x 10 .ε x x ε ε
For ε = 0.001 this becom∫es I = 999.9. The varia[nce bec]omes huge since〈 〉 1 10 1 1 10 1
f 2 = (f(x))2 dx = − ≈
V ε 10− ε 3x3 ε 30 3
.
ε
So, the variance is approximat〈ely 〉 1 1
σ2(f) = f 2 − 〈f〉2 ≈ 30 3 −ε 100 .ε2
This leads to a problem using a MC integration to solve this integral. Using a
sample xi on the interval [ε, 10] only one of ten parts will cover the dominating
107
Chapter 6 - Simulation of the Electro-Disintegration Reaction 16O(e, e′α)12C
area below 1 (see Fig. 6.2). In Table 6.2 the results of such an integration can
be found. As we can see, the error given is very large, and even with 10 000 000
generated events, the result is incorrect by more than 1%.
Now, we will use importance sampling to improve the variance and thus the MC
integration. For this purpose, we choose p(x) = 12 = f(x), with P (x) = − 1 .x x
Then the integral becomes∫ 1/ε
= d = vol([0 1 1 ]) = 1 1I P . , /ε −
0.1 ε 10
,
and the variance becomes ( )
2 fσ = σ2(1) = 0.
p
This is a perfect case, since only one sampling point is sufficient now to calculate
the integral to get the true value. This example is only intended to illustrate the
functionality of importance sampling. In practice, the MC integration is used to
solve integrals that cannot be solved analytically. Thus the choice p(x) = f(x) is
not possible, because otherwise the integral could be solved without MC. In this
case, a function p(x) should be chosen which describes the form of f(x) as close
as possible.
2
f(x) = 1
x2
1.5
1
0.5
0
0 2 4 6 8 10
Figure 6.2: The function f(x) = 12 diverges for small x. A sample in thex
interval [ε, 10] sufficiently covers the range above 1, but not the range below.
This leads to a problem when calculating the integral with the MC integration.
6.1.2 General Idea of a Cross Section Simulation for a Scattering Ex-
periment
In this section we will take a close look on the general idea how to simulate the
cross section for scattering experiments using the MC integration. The goal of
such a simulation is for example to estimate the count rate for a specific detector
or to determine the expected statistical errors for a dedicated experiment. In all
these cases an integration of the differential cross section is needed. However, we
are not looking for the total cross section, but for the cross section in small areas,
e.g., in a specific window of a detector∫. The cross section of a specific set Ωi ⊂ Ωcan be calculated with
σi :=
dσ dΩ .
Ωi dΩ
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6.1 Overview of Physics Simulations
Table 6.2: This table shows the results of calculating the integral of the function f(x) =
1
2 over the interval [ε, 10]. The true value is I = 999.9 for ε = 0.001. Even for 10x
million sample points, the result of the MC integration is deviates by more than 1%.
The calculated error for MC is large because the variance is large. Therefore, to better
calculate the integral, the variance must be reduced. One method for this is importance
sampling.
N I errcalc ∆rel [%]
1000 198.42 25 634.80 80.16
10000 257.07 43 503.59 74.29
100000 826.12 897 671.77 17.38
1000000 851.14 258 797.36 14.88
10000000 982.92 97 977.15 1.70
Since the geometrical description of a detector area can be difficult, this formula
is modified in a more general form using the acceptance function A(Ω). This
describes the efficiency, 0 for not det∫ected and 1 for full acceptance:
:= dσσA dΩA(Ω) dΩ . (6.3)Ω
This formula allows us to calculate the cross section in any possible detector setup.
Only the acceptance needs to be defined properly. To calculate the exptected
number of events N for a given d∫ete∫ctor setup within a time interval I we can use
= dσNA dΩA(Ω)L dΩ dt ,I Ω
where L is the luminosity. F∫or time indepen∫dent reactions this becomes
= dσNA dΩA(Ω) dΩ · L dt = σ · L.∫ AΩ I
The part L := I L dt is called integrated luminosity. For simplicity, we assume
that the luminosity is constant, so we get L = L · I. To determine the count rate,
the integral in Eq. (6.3) is estimated using a MC integration.
As explained in Section 6.1.1 the integral Eq. (6.3) can be approximated as
∫ ∑N
σA =
dσ (Ω) dΩ ≈ vol(Ω) dσdΩA dΩ(ωi) · A(ωi), (6.4)Ω N i=1
for a set ωi ∈ Ω of randomly generated events. To perform such a MC integration
different parts need to be defined. The first step is to create a random set of
events ωi ∈ Ω through an event generator. This can be divided into two parts,
the initial state particle generator and the final state particle generator.
6.1.3 Initial State Particle Generator
Each reaction can be described using a Feynman diagram. The initial state par-
ticle generator will produce all incoming particles of such a Feynman diagram.
Random numbers may be included at this point. For example, to randomly gen-
erate the starting point, direction or momentum of a particle. These values can
be distributed differently, e.g. homogeneous, Gaussian, linear or Poisson. For
example, in a scattering process the beam position may be Gaussian distributed.
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Chapter 6 - Simulation of the Electro-Disintegration Reaction 16O(e, e′α)12C
For a fixed target of length d and homogeneous density, the target particle can
then be generated along the path of the scattering particle in the range [−d/2, d/2].
How the initial particle generator looks like in detail depends on the experimental
setup and the simulation itself.
6.1.4 Final State Particle Generator
After the initial particles are generated, the final state generator generates the
parameter of the outgoing particles of the Feynman diagram. They are generated
according to the Feynman rules and in agreement with the conservation laws. The
reaction plays a decisive role, since it determines possible values for the outgoing
particles. The integral or the integration ranges determine which parameters must
be generated randomly. The number of different random parameters corresponds
to the number of degrees of freedom of the reaction. If other parameters should
be generated randomly, a corresponding variable transformation must be applied
to the integral. This can be useful for various reasons and must be adapted
to the problem. It is important to mention that the random numbers must be
independent. Therefore, such transformations must be performed carefully. At
this point, one would then also include importance sampling. The weight wi of
the event is calculated as
wi = |det(J(ωi))| · V,
with J the Jacobian matrix of the variable transformations and the volume ele-
ment V . The volume element of the integral results from the parameter ranges
which are used to generate the random numbers.
In the example of elastic scattering there are only two degrees of freedom, the
scattering angle θ as well as the rotation angle φ. From there all other parameters
of the outgoing particles can be calculated. Instead of generating θ ∈ [0, π],
it is more efficient to generate the value cos(θ) ∈ [−1, 1] through the variable
transformation sin(θ) dθ = d cos θ on the integral. Furthermore, we have [Pov+15]
d ∝ 1σ ( )
sin4 θe2
for the cross section of elastic scattering. Thus we know that small scattering
angles are preferred. According to importance sampling a transformation
p(sin2(x)) = 1sin4(x)
reduces the variance and improves the effectiveness of the MC integration.
To further increase the effectiveness of the MC integration, it is important to
limit the range of randomly generated parameters in advance. This is to avoid
generating values in ranges that are not relevant to the integration. For example,
if it is known that a particular detector can only detect particles in a certain
parameter range, the integration volume should be limited to this range to save
expensive computation time.
6.1.5 Physics Model Calculations
After the events have been created it is necessary to calculate the physical model.
At this point the value dσdΩ(ωi) is calculated by using the values generated in
the event generator. If the differential cross section is known, it can easily be
110
6.1 Overview of Physics Simulations
calculated. However, in most cases this will not be the case. Therefore, it is
necessary to use an adequate model to calculate the differential cross section.
At this point, for example, a simplified model can be used. However, it is also
possible to approximate the differential cross section from values obtained from
fitting existing data. The user is free to develop a suitable model. It is only
important that the value of the differential cross section can be calculated for
the respective events. This part is probably going to change and evolve over the
duration of an experiment in order to improve the MC integration result and thus
the desired predictions. With the physics model calculation, the weight of the
event is expanded as follows:
wi = |det(J( ))| · ·
dσ
ωi V dΩ(ωi).
6.1.6 Acceptance Simulation
In the final step to calculate the weight, the detectors become a part of the
simulation. For each event, it must be calculated whether it can be detected by
the detector system. If the event is not detected, the function A(ωi) takes the value
0. If an event is detected with absolute certainty, the value is 1. By characterizing
the detectors, the efficiency of the detectors can be determined and the acceptance
function can be adjusted accordingly. How the acceptance function is calculated
in detail depends strongly on the detector system and its design. Simple models
may be designed to geometrically calculate whether the detector area is traversed
by the particle, and output the value 1 if it is. However, more complex models
are also possible. For example, the track of a particle in a spectrometer could
be followed to calculate the focal plane location. Then, the acceptance could be
calculated, taking into account the efficiency of the focal plane detector. It is also
possible to combine several detector systems into one system using logical AND
or OR operations. This would correspond to a detector array or a coincidence
for example. No matter how the acceptance is simulated, in the end the function
A(ωi) results in a value in the interval [0, 1] which corresponds to the probability
that an event is detected by the system. The acceptance also contribute to the
weight of the event:
wi = |det(J(ωi))| · ·
dσ
V dΩ(ωi) · A(ωi).
6.1.7 Calculation of the Cross Section
After all events and weights are calculated they should be stored in a dedicated
way. It is advisable to choose a data format that is designed for large amounts of
data. For example, one can use the ROOT library developed by CERN to store
the data in a tree structure [ROOT]. This can later be read and processed by the
user’s own analysis or by an analysis based on ROOT.
The total cross section can finally be calculated as
= 1
∑N
σA wi.
N i=1
At this point, however, it is usually not the total cross section that is of inter-
est, but rather how the cross section is distributed over small parameter ranges.
Histograms can be created out of the events and the weights for example. Each
bin of a histogram then corresponds to the integral over a small parameter range
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Chapter 6 - Simulation of the Electro-Disintegration Reaction 16O(e, e′α)12C
and thus this represents the integrated differential cross section over this range.
By multiplying the luminosity to each bin the expected count rates is given. The
extent of the analysis depends heavily on what information a user would like to
receive from the events and the associated weights.
6.2 Implementation of the S-Factor Simulation
In this section we will take a closer look at the simulation of the measurement of
the electro-disintegration reaction 16O(e, e′α)12C at MAGIX. We will focus on the
Phases 1 and 2 of the measurement.
For the simulation a C++ framework developed in the MAGIX group was used.
Some parts like the initial state generator and an acceptance simulation of the
spectrometer were already included in the framework. All other parts were devel-
oped and integrated into the system during this work, especially the final state
generator and the model. In addition, I participated in the general development
of this software. Besides the development of a physics library based on the one
of ROOT, I also worked on the generators and the acceptance classes. Since the
description of all this work would go beyond the scope of this thesis, we will limit
ourselves here to the development of the simulation for the S-factor and only
describe its most important points.
When creating simulations, a consistent coordinate and unit system must be spec-
ified. In Table B.9 the coordinate system of the target and the generator system
used for the simulation is specified. In addition, the base units defined in Ta-
ble B.10 are used in the simulation software.
6.2.1 Overview of the Initial State Generator
As mentioned before, some initial state generators already existed in the MAGIX
framework. For the generation of the beam particle the ideal beam generator was
used. As the name implies, the beam particles are ideally generated, i.e. no param-
eters are randomly assigned. The mass and the charge of the produced particle is
loaded from an appropriate particle list using the MC particle numbering scheme,
also known as particle identification (PID), [Zyl+20, section 44]. In a next step a
starting point is associated to the particle. Since the beam particles are assumed
as ideal and thus there is no deviation from the ideal beam, we have x = y = 0.
Then, the beam energy and the direction are defined as a four-momentum. For
example an electron (PID 11) with a beam energy of 105MeV will have the mass
me, charge 1, together with the starting point ~x0 and a four-momentum Kµ as:
 
 
=  00  
105
0 
~x  , Kµ0 =  0
−∞ √ 

.
1052 −m2e
The −∞ indicates internally that the particle is generated in the source.
To generate the target particle a homogeneous cube generator was used. A uniform
particle distribution in a 3-dimensional box with widths dx dy and dz around the
origin is assumed. In the case of the simulation for the S-factor, the widths
dx=dy=0 and dz=2 are used. In addition, the generator assumes that the target
particles are at rest. Similar to the beam generator, the mass and the charge
of the particle are loaded using a MC particle numbering scheme. For example
112
6.2 Implementation of the S-Factor Simulation
an oxygen target particle (PID 1000080160) will have the mass M16 with chargeO
Z = 8, together with the starting point ~x0 and a four-momentum P
µ
 16 as:O
= 
 
0 M 160 OPµ =  0~x0 , 16 0 

,
O 
z 0
where z ∈ [−1, 1] is randomly generated.
6.2.2 Development of the Two-Body Final State Generator
The outgoing particles: the scattered electron, the recoiled α-particle and the
12C particle have to be generated in the final state generator. The differential
cross section of the electro-disintegration reaction 16O(e, e′α)12C has five degrees
of freedom Eq. (3.39):
dσ
dωdΩ′edΩcms
,
α
where ω = Ee − E ′e, dΩ′e = d cos(θe) dφe and dΩcmsα = d cos(θcmsα ) dφcmsα . It is easy
to see that dω = dE ′e. For the simulation we split the Feynman diagram Fig. 3.1
and thus the calculations into two parts, see Fig. 6.3.
Kµ(E ,~e k) K′µ(E′e,~k ′)
e− e− P
µ
12 (E , p~ )C 12C 12C
+ 16O∗ 12Cγ∗
Pµ∗ (E∗, p~∗) α
16O 16O∗ Pµα (Eα, p~α)
Pµ16 (E , p~ ) P
µ(E , p~ )
O 16O 16O ∗ ∗ ∗
Figure 6.3: The Feynman diagram of the electro-disintegration reaction of 16O
(Fig. 3.1) can be separated into two parts. The left diagram represents the
formation of the compound nucleus state 16O∗. The right one represents the
decay into α and 12C.
Instead of the two parameters E ′e and cos(θe), the invariant parameters W and q2
are used for the MC integr(ation. T)hus a(var)iable transformation Φ
E ′Φ : ecos( ) 7→
W ′
2 (Ee, cos(θθ q e)),e
is necessary. In Section 3.8 w√e find
W = √2ωM16 +M2 2O 16 + q (6.5)O
= 2(Ee − E ′e)M16 +M216 + q2,O O
where the four-momentum transfer is given as Eq. (3.4)
q2 := 2m2e − 2EeE ′e + 2k√k ′ cos θe √ (6.6)
= 2m2e − 2EeE ′e + 2 E2e −me E ′ 2e −me cos θe.
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Chapter 6 - Simulation of the Electro-Disintegration Reaction 16O(e, e′α)12C
To determine the Jacobian matrix JΦ the partial derivatives of the transformation
have to be calculated
∂W M
′ = −
16O ∂W
∂Ee W ∂ cos(θe)
= 0
∂q2 2
′ = −2
k ∂q
E + 2E ′e · cos(θe) = 2kk ′.
∂E e ′e k ∂ cos(θe)
This results in
2M16 kk ′det(J ) = − OΦ .
W
The weights for the eve∣nts (are t)h∣en defined as
= ∣∣w det J−1 ∣∣Φ · vol( 1V ) = |det(JΦ) · vol(V )|
= W2 · vol(V ),M16 kk ′O
with the integration volume ∫
vol(V ) = dW dq2 dφ dΩcmse α .
Since the differential cross section is pr(oport)ional to the Mott cross section wehave
dσ dσ 1
d dΩ′ dΩcms ∝ω dΩ′ ∝ q4 .e α e Mott
So, events with low q2 are preferred. This requires a further variable transforma-
tion according to importance sampling. Therefore we choose
( 2) := 1p q (q2)2 .
Then we get dq2 = q4 dP with P ∈ {1/q2}. Finally, the weight is given as
W
w = q4 · ′2 ′ · vol(V ).M16 kkO
In the case o∫f an connected volume V ′ the integration volume becomes
vol(V ′) = ∫ dW dP dφ dΩ
cms
∫ e αW P ∫ φ ∫max max e,max cos(θcmsα,min) ∫ φcmsα,max= dW dP dφe d cos(θ cmsα) dφα .
Wmin Pmin φ min cos(θcms cmse, α,max) φα,min
With ∫ P ∫max q2max
ν := dP = p(q2) dq2 = 1 1
2 q2
− 2 ,Pmin qmin min qmax
we get ( ) ( )
vol(V ′) = ν · (W −W ) · (φ cms cms cms cmsmax min e,max − φe,min) · (cos θα,min − cos θα,max ) · (φmax − φmin).
For each event the parameters are generated as a random value. For example,
if uW is a random number in the range [0, 1] the invariant mass of an event is
calculated as
W = Wmin + uW · (Wmax −Wmin).
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6.2 Implementation of the S-Factor Simulation
In the case of four-momentum transfer the value is calculated as
q2 = 11 ,
2 − u 2 · νq qmin
with a random number uq2 ∈ [0, 1].
With the set (W, q2, φ , cos(θcms), φcmse α α ) each four-momenta of the outgoing parti-
cles can be determined. In addition, a starting point is defined for each particle.
The charge and the mass of each particle is loaded from a particle list using the
PID. All this information is needed in the further steps of the simulation.
The starting point for all outgoing particles will be defined as the interaction point
from the beam particle with the target particle. In this particular simulation this
is equal to the starting point of the target particle xi = x0.
The four-momentum of the electron K′µ is defined by (E ′, cos(θe), φe). The values
E ′ and cos(θe) are calculated by using the formulas Eqs. (6.5) and (6.6).
In the cms the intermediate state Pµ∗ (E∗(, p~∗))is given as
Pµ∗ =
W
~0 .
To calculate the four-momentum of α and 12C we can use the formulas of the
two-body decay from [Zyl+20, chapter 48]:
2 2 2
cms W −M= 12
+M
C αEα 2W
Ecms cms12 = W − EC √( α )( )
W 2 − (M +M )2 W 2 − (M −M )212C α 12C α
pcms cmsα = p12 =C 2 .W
In the cms we have p~α = −p~12 with a homogeneous angular distribution. InC
this particular simulation the range used for the cms particle generation is set to
cos(θcmsα ) ∈ [−1, 1] and φcmsα ∈ (−π, π]. In the final step the four-momentum of
the α-particle and the 12C is boosted into the lab frame.
At the end of the final state generator the four-momenta K′µ,Pµα and Pµ12 in theC
lab frame, the interaction point ~xi and the weight w is calculated for each event.
This information is passed to the model calculation in combination with the PID
of each particle.
6.2.3 Development of the S-Factor Model Calculation
The cross section Eq. (3.39) must be determined for each event in the model calcu-
lator. Some assumptions and simpl∑ifications are made to calculate the proportion
ṽKR
cms
K .
K
Since the four-momentum transfer Q2 will be small (O(10−4 GeV2) in Phase 1;
O(10−9 GeV2) in Phase 2), the virtual photon is approximated as quasi-real.
Therefore, the longitudinal and both interference parts are neglected in the current
version of the simulation:
ṽL = ṽL = ṽL = 0.
115
Chapter 6 - Simulation of the Electro-Disintegration Reaction 16O(e, e′α)12C
So, only the transverse part is calculated. The same assumption is made to
determine the nuclear response function RcmsT , so that Eqs. (3.54) to (3.56) are used
to calculate the value of RcmsT . At this point, the value of SE1(Ecms) and S (Ecmsα E2 α )
are also taken into consideration. For this purpose, the AZURE2 software [Ud15]
was used to extrapolate the S-factor of the ground state transition for E1 and E2
in the cms energy range W = 0.01 − 20 MeV. Therefore, the AZURE2 software
was configured with the parameters of the supplementary material provided by
[deB+17]. The output of AZURE2 is stored in a file containing the cms energy
and the value of SE1 or SE2. In the extrapolation, different step widths in different
energy ranges were chosen (see Table B.11). The produced files of the AZURE2
software are loaded into the model calculator. To calculate the S-factor value for a
given Ecmsα , the calculator searches for the two entries which cms energy is closest.
The value of the S-factor is calculated by a linear approximation based on the two
entries. Since the loaded list is sorted, the value can be determined quickly.
Finally, the weight is calculated as
w = q4 · W2 ′ · vol(
′) · dσV
M16 kk dωdΩ′edΩcms
,
O α
with
dσ M ′2αM12C βe
d dΩ′ dΩcms = 8 3 · · p
cmsσ · ṽ Rcmsα Mott T T ,ω e α π W βe
and 
~c · pcms ·W · e−2πη 
Rcms = α · S [P (cos(θcmsT 2 cms E1 0 ))− P2(cos(θ
cms))]
α · Eγ ·[MαM α α12C · Ecms α ]
+ S cmsE2 P0(cos(θα )) +
5
P (cos(θcms))− 127 2 α 7 P4(cos(θ
cms
α ))
6 √ + √ cos (φ (E)) S S [P (cos(θcms))− P (cos(θcms))] .
5 12 E1 E2 1 α 3 α 
The phase shift was approximated equal to Eq. (2.30) as
cos (φ12(E)) = √
2
η2
.
+ 4
6.2.4 Implementation of Acceptance Simulation
After generating an event and calculating the weight, the value of the acceptance
function A must be determined. Therefore, the detector setup must be simulated.
Each detector is implemented as an ideal version in the MAGIX framework as a
separate C++ class. The acceptance function of each detector returns the value 1
if the particle is detected. Otherwise it returns the value 0.
In this section we will discuss how the different detector systems are integrated
into the simulation. Each detector is individually designed to suit the specific
requirements. After a review of the individual detectors for the electrons and the
α-particles, the linking of several detectors as a set or as coincidence is discussed.
116
6.2 Implementation of the S-Factor Simulation
Spectrometer
The acceptance for the spectrometers was already implemented to the frame-
work. The spectrometer acceptance is a 4-dimensional space with the parameters
k, θ, φ, y in the spectrometer coordinate system. The spectrometer acceptance
was implemented as a binary tree as discussed in [Mül14]. Each level of the tree
represents a bisection of a coordinate and one moves to the next level after it-
erating over all coordinates alternatingly. Each evaluation represents a bisection
of the available phase space in that coordinate, so that after iterating over all
coordinates, the search restricts the phase space of each node to an hypercube of
dimension n and volume 12 compared to the previous iteration. Any leaf that isn
completely inside or outside the acceptance space is not subdivided further. This
results in different volumes for each leaf in the different iteration steps. The result
of the iterations indicates whether the particle hits the focal plane. In this case
the acceptance returns A = 1, otherwise A = 0.
Zero Degree Tagger
The zero degree tagger is implemented as a purely geometric detector in the sim-
ulation software. Each particle generated in the final state generator is passed to
the acceptance class of the zero degree tagger to test whether any of these particles
will be detected. For each particle, the acceptance is calculated separately. First,
the detector verifies the charge of the particle. This must be negative. The next
step is to confirm that the particle’s momentum is within the momentum accep-
tance range. It is assumed that any electron with a scattering angle below θmax is
focused by the first deflecting magnet onto the active area of the detector system.
Thus, the last step is to prove that the scattering angle is less than θmax. If one
particle passes all three tests, the zero degree detector returns A = 1, otherwise
A = 0. The accepted momentum range as well as the maximum angle acceptance
can be configured in the initialization phase of the simulation.
Silicon Strip Detector
A silicon strip detector array is used for the detection of the α-particles. This
array consists of a number of individual configured silicon strip detectors. Similar
to the acceptance simulation for the zero degree tagger, the simulation of such a
silicon strip detector is purely geometric. Each silicon strip detector is configured
with a set of parameters that can be specified in a configuration file that is loaded
into the simulation during the initialization phase (see code D.3 and D.4). There,
the minimum accepted particle energy, the charge sensitivity, the rotation angle,
the distance to the target center dS, and the length lS as well as the width wS of
the detector can be specified. In addition, each detector can be restricted to a list
of particles. For this purpose, a list of PIDs is used to configure the detector.
For each particle calculated in the final state generator, the acceptance class of
the silicon strip detector is used to verify whether it will be detected. If a PID
list is set, it is first checked whether the particle’s PID is listed there. If not, it
will not be accepted by the detector. In the next step, the energy and the charge
are tested. In other words, it is checked if the energy of the particle is above the
minimum accepted energy value and if the charge of the particle is positive. In
the last step, it is proved whether the particle hits the detector area. For this
purpose, a straight line is formed with the interaction point as the starting point
and a direction parallel to the three-momentum of the particle. The entire system
is then transformed so that the detector is located in the x-y plane with z = dS.
117
Chapter 6 - Simulation of the Electro-Disintegration Reaction 16O(e, e′α)12C
Then, the x and y positions of the line are determined for z = dS. In the case
|x| < wS/2 and |y| < lS/2 the particle hits the detector. If one of the particles hits
the detector, the acceptance class makes A = 1, otherwise A = 0.
Acceptance Set and Coincidence
To simulate a set of detectors or the coincidence of two detector systems, addi-
tional acceptance functions are implemented into the simulation framework. The
acceptance for a set of detectors is calcu∏lated as follows
A = 1− (1− Ai).
i
Where Ai are the individual acceptances of the detectors. In the current version
of the simulation, this gives A = 1 when even one detector can accept one of the
particles. This was used for the simulation of the silicon strip detector array.
The acceptance for a coincidence of sev∏eral detectors, on the other hand, is cal-culated as
A = Ai,
i
where Ai are the individual acceptances of the detectors involved. In the cur-
rent version of the simulation, this gives A = 0 when even one detector cannot
accept any of the particles. This was used to simulate coincidence between the
spectrometer (or zero degree tagger) and the silicon strip detector array.
6.2.5 Analysis: Calculation of Expected Count Rate
After an event is generated, the weight for this event is estimated, and the ac-
ceptance for the event is determined, the event is passed to the analysis. This
processes the data depending on the information to be obtained from the simu-
lated data.
In general, the result of the MC integration gives the total integrated cross section
for a given acceptance A.
1 ∑N
σA = wi,
N i=1
where wi denotes the weight for each event, and N the number of total events.
This is only one single value and offers rather less information. The basic idea is to
split the integration space into small volumes where the differential cross section
can be assumed to be constant. In the one-dimensional case such an integration
can be written for example∫asE+∆E
( ) = dσ ( ) d ≈ dσσ E d E E d (E) ·∆E.E E E
If a parameter is entered into a histogram together with the associated weight for
each event, a weighted histogram is obtained. After all events have been written
into the histogram, it must be scaled by the factor 1/N. In this particular case,
each bin represents the integrated differential cross section, integrated over the
parameter range on which that bin is based. An example can be found in Fig. 6.4.
There, the cms energy of the α − 12C system is entered in a one-dimensional
histogram with the associated weight for each event. The left y-axis shows the
value of each weighted bin and represents the cross section on a small energy width
∆E = 0.069 MeV. The right y-axis of this histogram represents the number of
118
6.2 Implementation of the S-Factor Simulation
expected reactions per energy width ∆E. The number of reactionsNr is calculated
using
Nr = σ · L,
with the integrated luminosity
L = L · I,
where a constant luminosity L = 1034 cm−2 s−1 integrated over the time duration
T = 4 weeks = 2 419 200 s was assumed.
10−4 1012
10−6 1010
10−8 108
10−10 106
10−12 104
10−14 102
10−16 100
10−18 10−2
0EG 1 2 3 4 5 6 7
Ecms[MeV]
Figure 6.4: Histogram representing the weighted cms energy Ecms for Phase
2. Each bin represents the integration over a small energy width of ∆E =
0.069 MeV. The left y-axis represents the cross section while the right y-axis
represents the expected number of reactions.
Also multi-dimensional histograms can be produced. Figure 6.5 shows a two-
dimensional histogram representing the hitmap for the silicon strip detector array
of Phase 1. The histogram is scaled by the same integrated luminosity and repre-
sents the expected number of detected events in coincidence with the spectrometer.
Some of the histograms are produced through the simulation and are stored in a
ROOT file so that they are available after the simulation [ROOT]. All histograms
can be configured in the initialization phase of the simulation, for example to
set the parameter range or the number of bins (see code D.2). The analysis is
started directly with the simulation and processes the events as soon as they were
generated. This has the advantage that the results are available directly after the
simulation in the form of histograms and can be used for further processing. How-
ever, one disadvantage is that the raw data of the events and also the weights are
not available after the simulation and cannot be regenerated from the histograms.
This means if changes are made to the analysis, the entire simulation has to re-run
to store the desired information. Depending on the simulation and the number of
events generated, this can take a long time.
119
σ[b]
N
Chapter 6 - Simulation of the Electro-Disintegration Reaction 16O(e, e′α)12C
5
3 10
2 104
1
103
0
102
−1
−2 10
−3
1 1− −0.5 0 0.5 1
Cos(θ)
Figure 6.5: Hitmap for the silicon strip detector array. It shows the expected
number of events detected on the surface of the different silicon strip detectors
for one possible configuraton in Phase 1.
Therefore, all data generated in the simulation was additionally stored in a file.
Thus it is available for later analyses. For this purpose, all data that can be
obtained from the events are stored in a tree structure. Therefore, TTree from
the ROOT package was used [ROOT]. A separate branch is created for each
particle so that the information can be stored and retrieved individually. The
structure for all particle branches is identical and contains the vertex, the mass,
the four-momentum and the PID of the particle. In this particular case, the vertex
corresponds to the interaction point and is therefore identical for all particles. In
addition to the initial and final state particles, the intermediate state particles,
γ∗ and 16O∗, were also stored in the ROOT file. Additionally, to the particle in-
formation, a further branch containing the event-related data is created. Besides
the calculated weight, the parameters randomly generated in the final state gen-
erator are also stored in this branch. The explicit specification of the two TTree
structure used can be found in code D.1.
The produced ROOT TTree was used to perform more complex analysis on the
data, such as calculating the exptected statistical errors for the measurement of
the E1 and E2 ground state distribution of the S-factor measurement. This was
done in several steps.
In the first step the data is splitted into small cms energy ranges accordingly.
For example the cms energy range [0.3 MeV, 3 MeV] was splitted in smaller ranges
[Ei, Ei+1) with E0 = 0.3 MeV and Ei+1 = Ei + ∆E for ∆E = 0.05 MeV. For each
energy range a histogram was produced entering cos(θcmsα ) with the coresponding
weights. To get the distribution of expected detected events over the cms angle
of the α-particle, each histogram was scaled with the factor L/N, with N beeing
the total number of generated events. The number of particles detected in a time
intervall T with time independent particle flux and time independent detector
efficiency is Poisson distributed [Lyo86]. Therefore, the error for each bin k was
approximated by using the formula√
Nk + 2, (6.7)
120
Φ [rad]
6.2 Implementation of the S-Factor Simulation
where Nk is the number of events of bin k. The +2 in the square root is used to
also include the empty bars. Figure 6.6 shows an example of a histogram including
the approximated errors for Phase 2.
40000
35000
30000
25000
20000
15000
10000
5000
0
−1 −0.5 0 0.5 1
cos(θcmsα )
Figure 6.6: Number of expected events distributed over the cms angular of the
α-particle in the cms ene√rgy range Ecms = 1.3 − 1.4 MeV. The error of each
bin k is calculated using Nk + 2 where Nk is the number of events of bin k.
In addition one histogram representing the isotropic generated value cos(θcmsα )
using weight w = 1 was produced. For this histogram the data of the entire
energy range was taken into account. This histogram is also scaled by L/N and
the error for each bin was calculated using Eq. (6.7). Figure 6.6 shows an example
of this histogram including the approximated errors. Even if the cms angle of the
α-particle is isotropically generated the result of this histogram looks different.
This is due to the acceptance, as the projection of the silicon strip detectors on
the surface of the unit sphere is not conformal. In Fig. 6.5 one can already see
that for the assumed detector setup the acceptance in the range close to ±1 for
cos(θcmsα ) is higher.
In the second step the statistical error of the MC integration must be eliminated.
Therefore, each histogram is normalized by the isotropic histogram. Technically,
this means that the value of each bin of a histogram is divided by the corresponding
value of the bin of the isotropic histogram. The errors are calculated by simple
error propagation. As a result, for each cms energy range, one obtains a histogram
representing the distribution of the cross section σ for the cms angle of the α-
particle. An example of one of these produced histograms can be found in Fig. 6.8.
In the last step each of the produced normalized histograms were fitted using
Eq. (2.31) with the fit parameters a := σ σE1 and b := E2/σE1 as discussed in Sec-
tion 2.3.1. The phase shift was calculated using Eq. (2.30). The result of such a
2-parameter fit can be found in Fig. 6.8. As result of each fit one gets the values
σE1(Ei) and σE2(Ei) and the corresponding errors ∆σE1(Ei) and ∆σE2(Ei). The
error ∆σE1(Ei) is equal to the error ∆a of the fit parameter. The error for E2 was
calculated using error propagation
∆σE2(Ei) = σE1(Ei) ·∆b+ b ·∆σE1(Ei).
The errors correspond to the statistical errors that are expected for the measure-
ment with an identical detector setup. Figure 6.9 shows the results of this analysis
121
Nr
Chapter 6 - Simulation of the Electro-Disintegration Reaction 16O(e, e′α)12C
7·1013
6·1013
5·1013
4·1013
3·1013
2·1013
1·1013
0
−1 −0.5 0 0.5 1
cos(θcmsα )
Figure 6.7: The histogram represents the isotropically distri√buted cms angle of
the α-particle. The error of each bin k is calculated using Nk + 2 where Nk
is the number of events of bin k.
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
−1 −0.5 0 0.5 1
cos(θcmsα )
Figure 6.8: The histogram represents the weighted cms angle of the α-particle
for the energy range Ecms = 1.3− 1.4 MeV corresponding to the cross section
distribution. The error of the MC integration is elimated as described in the
text. The data is fitted using Eq. (2.31).
for the different Phases 1 and 2 for one specific detector setup. To cross-check
the validity of the simulation results, an independent simulation was performed
within the MAGIX group based on the theoretical assumptions and calculations
described in this thesis. It confirmed the results, that will be discussed in the next
section.
6.2.6 Discussion of the Simulations Results
In this section we will discuss the results of the simulations performed as part of
this work. A summary of the parameters used for the simulations and the analysis
of the electro-disintegration measurement at MAGIX for Phases 1 and 2 are given
in Table 6.3. The results of the simulations can be found in Fig. 6.9.
122
σ [nb]
Nisotropic
6.2 Implementation of the S-Factor Simulation
Table 6.3: Summary of the relevant simulation and analysis parameters.
Parameter Value
Electron Beam1,2 energy 105MeV
current 1mA
Oxygen Target1,2 areal number density 2× 1018 particles/cm2
gas type pure 16O
Spectrometer1 central momentum 75 MeV
central in-plane angle 13°
Zero Degree Tagger2 energy acceptance E′e ≥ 7 MeV
angular acceptance θe ≤ 0.5°
Silicon-Strip Detector Array1,2 detector count 5
active detector area 50× 50 mm2
distance 10 cm
central in-plane angle 30°, ±90°, ±120°
Luminosity1,2 1× 1034 cm−2 s−1
Time Period1,2 4weeks
Integrated Luminosity1,2 2.4192× 1040 cm−2
= 2.4192× 107 nb−1
Ecmsα -Range1,2 0− 25 MeV
1 MAGIX Phase 1
2 MAGIX Phase 2
In Figs. 6.9a and 6.9b, the data from previous measurements is given for the E1
and E2 ground state contributions. The E1 and E2 contribution are fitted using
the AZURE2 software [Azu+10; Ud15] which was configured with the parameters
from the supplementary material of Ref. [deB+17]. The results of the simulations
are added to these figures to show the expected errors compared to the existing
data. For this purpose, the S-factor values calculated with AZURE2 were added
to the figures with the calculated errors from the simulations. In Fig. 6.9a the sim-
ulated results for the E1 contribution are shown. As it can be seen, the expected
statistical errors for Phase 1 have a precision comparable to the existing data in
the range above 0.9MeV. A similar result is obtained for the E2 contribution
Fig. 6.9b. As expected, the error for the E2 contribution is larger than for the E1
contribution due to error propagation. Moreover, it can be seen that for Phase 2,
the expected errors in the region above 0.9MeV decrease greatly and, in addition,
it will be possible to study the region below. With this particular detector setup,
the E1 contribution can be determined down to 0.5MeV and the E2 contribution
down to 0.7MeV. For both components it will be possible to make contributions
in the range below 0.9MeV in Phase 2 and thus reducing the gap to the Gamow
peak.
The results of these simulations show that MAGIX can make significant measure-
ments of the S-factor. A reduction of the errors in E1 and E2 can be achieved by
MAGIX in two ways. The first important point is that reducing the errors in the
region above 1MeV allows a more accurate extrapolation of the S-factor to the
Gamow peak. This reduces the errors caused by the extrapolation. The second
point is that the simulations show that MAGIX can measure the S-factor down
to 0.5MeV in Phase 2. This will reduce the gap to the Gamow peak and thus
improve the extrapolation further.
123
Chapter 6 - Simulation of the Electro-Disintegration Reaction 16O(e, e′α)12C
103
1974 Dyer
1987 Redder
1988 Kremer
1996 Ouellet
1999 Roters
2001 Gialanella
102 2001 Kunz2004 Fey
2006 Assuncao
2009 Makii
2012 Plag
AZURE2 Fit
MAGIX Phase 1
101 MAGIX Phase 2
100
10−1
0 0.5 1 1.5 2 2.5 3 3.5
Ecms[MeV]
(a) E1 contribution data [DB74; Red+87; Kre+88; Oue+96; Rot+99; Gia+01;
Kun+01; Fey04; Ass+06; Mak+09; Pla+12].
103
102
101
100 1987 Redder1996 Ouellet
1999 Roters
2001 Kunz
2004 Fey
2006 Assuncao
10−1 2009 Makii2011 Schürmann
2012 Plag
AZURE2 Fit
MAGIX Phase 1
−2 MAGIX Phase 210
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Ecms[MeV]
(b) E2 contribution data [Red+87; Oue+96; Rot+99; Kun+01; Fey04;
Ass+06; Mak+09; Sch+11; Pla+12].
Figure 6.9: E1 (a) and E2 (b) contribution data fitted with the AZURE2 soft-
ware [Azu+10; Ud15] configured with parameters of the supplemental material
provided by [deB+17]. MAGIX Phase 1/2 shows the projected errors for the
Phases 1 and 2.
Although the longitudinal and the two interference terms were not included in
this simulation, the simulation provides a good estimate of the expected statis-
tical errors of the S-factor measurement at MAGIX, since the transveral term
dominates in the low Q2 range, whereas the omitted terms only play a minor
role, as discussed in Section 6.2.3. Compared to the photo-disintegration, the
124
SSEE22[[kkeeVV bb]] SSEE11[[kkeeVV bb]]
6.2 Implementation of the S-Factor Simulation
electro-disintegration contains more nuclear response functions. This means that
electro-disintegration has a potentially higher information content. This will al-
low a more accurate determination of the various contributions. In particular, the
error in E1 and E2 can be reduced. For more details, please refer to [FDM19].
At this point it is important to note that the results given are the results of
a simulation with one specific detector setup. The position of the silicon strip
detectors have been chosen in favor of the measurement of E1 (see Fig. 2.12). In
the experiment, several measurements will be performed with different detector
positions optimized for either E1 or E2. This will allow the reduction of the errors
of the E1 and E2 contribution best as possible
As mentioned at the beginning of the chapter, no simulation was performed to
determine the background. This is due to the fact that no final information on the
beam parameters of MESA or the efficiency of the detectors is currently available.
Compared to the direct measurements e.g. at LUNA-MV or Felsenkeller labora-
tory , the expected background from cosmic particles is low, due to the coincidence
measurement of the scattered electrons and the recoiled α-particles. This allows
MAGIX to perform the measurement next to an accelerator and not several hun-
dred meters under the ground. The detectors will be optimized for their respective
particle species. The amount of background signals in the final experiment and
how effectively they can be separated from the signals depends on the efficiency
of the detectors and other experimental conditions. Both are currently unknown
and must be determined experimentally.
One known background source are the signals arising from the contamination of
17O, 18O, and 15N in the oxygen gas (see Section 4.3.1). As described in Ref.
[FDM19], a ToF measurement can be used to separate the signals arising from
the contamination. To ensure this, the silicon strip detectors are additionally
optimized for high time resolution. Verifying whether it is possible to identify these
background reactions and determining the efficiency of the silicon strip detectors
will be part of the Phase 0 measurement at the A1 spectrometer facility at MAMI.
As calculated in the simulation we expect a very low event rate for cms energies
near the Gamow peak which makes it quite challenging to gain high statistics.
Nevertheless, MAGIX can still make a decisive contribution to the determination
of the S-factor. For cms energies above 0.9MeV, the expected counting rates are
high, so that in Phase 2 the statistical errors in this range will be low. This
allows to perform a more accurate extrapolation towards the Gamow peak and
consequently to reduce the error in the S-factor determination. Each additional
data point that can be taken in the sub-MeV scale improves the extrapolation.
Furthermore, the beam time period is a variable parameter. In addition the
integrated luminosity can be increased by the planned upgrade of the beam current
of MESA to 10mA. This means that statistical errors can be reduced due to longer
measuring times and a increased beam current.
125
126
CHAPTER 7
Summary and Outlook
The S-factor of the reaction 12C(α, γ)16O plays a decisive role in describing the
pre-supernova evolution of stars. As outlined in Section 2.5 ‘Production Factor’,
the S-factor is a crucial factor in the calculation of the isotope abundance ratio in
a star. Despite its importance and several attempts to determine the S-factor at
the Gamow peak EG = 0.3 MeV experimentally, the relative error of the S-factor
is currently greater than 17%. An overview and discussion of the already existing
data can be found in Section 2.4 ‘Results of Existing Measurements’. However, to
make reasonable predictions about pre-supernova evolution, a relative uncertainty
of under 10% is required. After considering all theoretical aspects as layed out in
Chapter 2 ‘The Reaction 12C(α, γ)16O in the Helium Burning Stage in Stars’, it
turned out that the problem in determining the S-factor is that the cross section
of the reaction 12C(α, γ)16O near the Gamow peak is dominated by the Coulomb
penetration factor. As a consequence, the cross section decreases exponentially
from ∼ 10−11 b at 1MeV to ∼ 10−17 b at 0.3MeV. This makes an experimental
determination of the S-factor at the Gamow peak so challenging. Thus, low-
energy high-precision experiments are needed to measure the S-factor near the
Gamow peak. In the coming years, several experiments are planned to measure
the S-factor using different methods. In Section 2.3 ‘Discussion of Measurement
Techniques’ an overview of different measurment techniques is given. Besides
direct measurements, e.g. planned at LUNA and Felsenkeller in Dresden, indirect
measurements via photo-disintegration, e.g. at the HIγS facility or at the ELI-NP
facility, are planned.
Another indirect measurement technique is the electro-disintegration measure-
ment of oxygen. For this purpose, electrons are scattered on oxygen and the
scattered electrons and the recoiled α-particles are detected in coincidence. Ac-
cording to the theoretical derivations made in Chapter 3 ‘Electro-Disintegration
Cross Section of 16O’ the E1 and E2 ground state contribution of the S-factor of
the reaction 12C(α, γ)16O can be determined from the electro-disintegration cross
section of oxygen.
This technique is applied for the planned experiment MAGIX. MAGIX will be
commissioned in the coming years at the high-intensity electron accelerator MESA
in Mainz, Germany, which is currently under construction. In addition to measur-
127
Chapter 7 - Summary and Outlook
ing the S-factor of the 12C(α, γ)16O, electromagnetic form factors or searching for
dark sector particles, and reaction studies of astrophysical relevance are planned.
The experimental setup of MAGIX at MESA for determinating the S-factor is
discussed in Chapter 4 ‘Accelerator and Experimental Setup’. This measurement
will be conducted in three phases (Phase 0, Phase 1 and Phase 2).
To estimate the expected statistical errors of the Phase 1 and 2 of the planned
electro-disintegration measurement at MAGIX a simluation was performed as part
of this thesis. Since in Phase 2 the scattering angle of the electrons will be less
than 0.5°, the cross section had to be derived without using the ERL. The deriva-
tions of the electro-disintegration cross section without using ERL is outlined in
Chapter 3 ‘Electro-Disintegration Cross Section of 16O’. Since information about
the efficiency of the detectors and the beam parameters is not yet available, no
background simulations were included. Furthermore, due to this reason, the effi-
ciency could not be included in the simulation of the acceptance of the detectors.
They are therefore determined as purely geometrically.
Based on the results of this simulation discussed in Chapter 6 ‘Simulation of the
Electro-Disintegration Reaction 16O(e, e′α)12C’, it can be stated that MAGIX will
be able to determine the S-factor in the low-energy range near the Gamow peak
of the helium burning stage EG = 0.3 MeV. Already in Phase 1, MAGIX will be
able to determine the S-factor in the cms energy range Ecmsα > 1 MeV with relative
errors competitive to previous experiments performed by other research groups.
In Phase 2, which includes the zero degree tagger, MAGIX can then reduce the
relative errors above 1MeV and perform high-precision measurements even on a
sub-MeV scale with a superior statistical significance.
The main conclusion from this thesis is that MAGIX can make an important
contribution to the determination of the S-factor in the coming years. Based on
these results, MAGIX will continue to plan and prepare measurements of the S-
factor. The simulation was only one of several steps that will contribute to the
success of the measurement of the S-factor for the reaction 12C(α, γ)16O.
Even though the simulation is promising, it has its limitations. Although back-
ground simulations could not be included, important conclusions can be drawn
from the results of this simulation. The most important aspects were discussed
in Section 6.2.6 ‘Discussion of the Simulations Results’. Despite the fact that
Phase 1 will not begin until 2023 at the earliest, it was very important to perform
this simulation at an early stage. The development of the detectors, especially
the silicon strip detector, is closely related to the results of this work. During
the development of the simulation, the analysis of the simulation was adjusted
to obtain important parameters needed for the development of the silicon strip
detector.
An essential next step is the experimental determination of the efficency of the
silicon strip detectors, especially for low cms energies. This will be part of Phase
0, in which the MAGIX gas jet target will be operated at the A1 facility at MAMI
in Mainz using oxygen. A futher important step in Phase 0 will be verifying the
ToF method to seperate the signals arising from the contamination of 17O, 18O,
and 15N in the oxygen gas. The information gained in Phase 0 will then in turn
be used to improve the simulation in particular including background calculations
and the efficency of the silicon strip detector.
Some MAGIX components, such as the gas jet target or the silicon strip detector
that will be used to detect the α-particles, are already tested in different labora-
128
tories. Another important part of this thesis was the development of the MAGIX
slow control system, which will later be used in the final experiment. As described
in Chapter 5 ‘Development of the MAGIX Slow Control System’ the slow control
system was extended by those test stands. To adapt the slow control system
based on the needs of MAGIX, the EPICS base software was extended by several
MAGIX modules that were build as part of this thesis. Running the test stands
with the MAGIX Slow Control system served as a stress test of the software. This
showed how reliable an EPICS-based slow control system can work even under
load. The structure of the EPICS-based MAGIX slow control was designed in
such a way that all devices that will later be needed for MAGIX can easily be
integrated. Special care was taken to ensure that the slow control can be used
in exactly the same way for the final experiment. To support users in getting
started with MXSlowControl and to provide professional tips for more complex
tasks to all team members a tutorial was created as part of this thesis. Under
consideration of the continous development of MXSlowControl this tutorial needs
to be updated constantly.
In summary, the simulation was a very important step to enable the measurement
of the S-factor of the 12C(α, γ)16O at MAGIX. In combination with the develop-
ment of the slow control for MAGIX as part of this thesis, MAGIX will be able to
prepare and perform Phase 0 up to Phase 2 based on the theoretical work carried
out in this thesis.
129
130
APPENDIX A
Response Functions
Following the derivation published in [RD89; FDM19] the nuclear response func-
tions RK in terms of the Legendre and the associated Legendre polynomials up
to octupole contrib(utions can be written as: )
RL = P0(cos θα)(| 2tC0| + | |2tC1 + |tC2|2 + | 2tC3| √√
+P1(cos
3
θα) 2 3|tC0||tC1| cos(δC1 − δC0) +) 4 5 |tC1||tC2| cos(δC2 − δC1)
(+√
18 |tC2||tC3| cos(δC3 − δ )35 C2
√
+P2(cos θα) 2| |2 +
10
√tC1 7 |
2
tC2| + 2 5|tC0||tC2| cos(δ)C2 − δC0)
+6 3( √7 |tC1||tC3| cos(δC3 − ) +
4 | |2δC1 3√tC3
+ (cos ) 6 3 8 7P3 θα 5 |tC1||tC2| cos(δC2 − δC1)) + 3 5 |tC2||tC3| cos(δC3 − δC2)√
(+2 7|tC0||tC3√| cos(δC3 − δC0) )
+P4(cos
18 2
θα)( 7 |√tC2| + 8
3
7 |tC1|| | cos( − ) +
18 2
tC3 δ)C3 δC1 11 |tC3|
+P5(cos )
20 5
θα ( 3 7 |tC)2||tC3| cos(δC3 − δC2)
+P6(cos
100 2
θα)( 33 |tC3| , )
RT = P0(cos )(| 2 2 2θα tE1| + |tE2| + |tE3| √ )
+P1(cos θα)(√
6 |tE1||tE2| cos(δ −
2
δ
5 E2 √ E1) + 12 35 |tE2||tE3| cos(δE3 − δ)E2)
+ (cos ) −| |2 + 5 | |2P2 θα ( tE1 7 tE2 + 6
2 | 27 tE1||tE3|√cos(δE3 − δE1) + |tE3| )
+P3(cos θα) −√
6 | || | cos( − ) + 2 14tE1 tE2 δE2 δE1 3 5 |tE2||tE3| cos(δ5 E3
− δE2)
131
Appendix A - Response Functions
( √ )
+P4(cos θα)(−
12 2 2 3 2
7 |√tE2| − 6 7 |tE1||tE3| cos(δE)3 − δE1) + 11 |tE3|
+ (cos ) −10 10P5 θα ( 3 7 |)tE2||tE3| cos(δE3 − δE2)
+ (cos ) −25{P6 θα ( 11 |
2
tE3| , √
R = P 1
√
TL 1 (cos θα) 2 3|
3
tC0||tE1| cos(δE1 − δC0)− 2√ 5 |tC2||tE1| cos(δC2 − δE1)
+√6√|tC1||tE2| cos(δC1 − δE2)− 6
3
) 35 |tC3||tE2| cos(δ5 C3 − δE2)
+(6
6
35 |tC2||tE3| cos(δC2 − δE3√)
+ 1(cos ) 2| || | cos( − ) + 2 5P2 θα tC1 tE1 δC1 δE1 3√|tC0||tE2| cos(δE2 − δC0)
+ 1√√
0 |tC2||tE2| cos(
3
δC2 − δE2)− 2 √7 |tC3||tE1| cos(δC3 − δ )7 3 E1 )
+4 2( √7 |tC1||
2 2
tE3| cos(δC1 − δE3) + 3 3 |tC3||tE3| cos(δC3 − δE3)
+P 13 (cos
3
θα) 2√5 |tC2||tE1| cos(δC2 −
4
δE1) + √√|tC1||tE2| cos(δC1 − δ5 E2)
+2 7√6 |tC0||
2 7
tE3| cos(δE3 − δC0) +)3 15 |tC3||tE2| cos(δC3 − δE2)
+2 7( 30 |tC2||tE3| cos(δC2 − δE3)√ √
+P 14 (cos )
6 3 3
θα 7 |tC2||tE2| cos(δC2 − δE2) + 2 √7 |tC3||tE1| cos(δC3 − δE1) )
+√6( |tC1||tE3| cos(δC1 −
6 3
δE3) + 11 2√|tC3||tE3| cos(δ − δ14 C3 E3) )
+P 15 (cos )
40 1 2
θα ( 3 √√ |t105 C3||tE2| cos(δC3 − δE2))}+ 10 105 |tC2||tE3| cos(δC2 − δE3)
+ 1(cos ) 50 2P6 θα 33 3 |tC3||tE3| cos(δC3 − δE3) · cos(φα)
RTT =−RT cos(2φα).
For the simulation, we used the nuclear response function where the Legendre
polynomials are expressed in terms of sine and cosine: [FDM19]
RL = | 2tC0| + 3| 2tC1| cos2(θα) +
5 2
16 |tC2| (1 + 3 cos(2θα))
2 + 7 264 |tC3| (3 cos(θα) + 5 cos(3
2
θα))
√ √
+ 2 3| 5tC0||tC1| cos(θα) cos(δC1 − δC0) + 2 |tC0||tC2|(1 + 3 cos(2θα)) cos(δC2 − δC0)√
+ 74 |tC0||tC3|(3 cos(θα) + 5 cos(3θα)) cos(δC3 − δC0)√
+ 154 |tC1||tC2|(5 cos(θα) + 3 cos(3θα)) cos(δC2 − δC1)√
+ 218 |tC1||tC3|(3 + 8 cos(2θα) + 5 cos(4θα)) cos(δC3 − δC1)√
+ 3532 |tC2||tC3|(30 cos(θα) + 19 cos(3θα) + 15 cos(5θα)) cos(δC3 − δC2),
132
Appendix A - Response Functions
= 3 | |2 sin2( ) + 15 | |2 sin2(2 ) + 21 | |2RT 2 tE1 θα 8 tE2 θα 256 tE3 (sin(2θα) + 5 sin(3
2
θα))
√
+ 3√52 |tE1||tE2|(sin(θα) sin(2θα)) cos(δE2 − δE1)
+ 3 5
( )
2√ 2 42 |tE1||tE3| sin (2θα)− sin (θα) cos(δE3 − δE1)
+ 3 35
( )
{4 2 |tE2||tE3| sin(2θα) sin(θα) + sin(3θα)− sin3(θα) cos(δE3 − δE2)
√ √
RTL = −√2 3|tC0||tE1| sin(θα) cos(δE1 − δC0)− 15|tC0||tE2| sin(2θα) cos(δE2 − δC0)
− 1 214 2 |tC0||tE3|(sin(θα) + 5 sin(3θα)) cos(δE3 − δC0)√
− 3|√tC1||tE1| sin(2θα) cos( 3 5δE1 − δC1)− 2 |tC1||tE2|(sin(θα) + sin(3θα)) cos(δE2 − δC1)
− 3 78 2 |tC1||tE3|(6 sin(2θα) + 5 sin(4θα)) cos(δE3 − δC1)√
+ 154 |tC2||tE1|(sin(θα)− 3 sin(3θα)) cos(δE1 − δC2)√
− 5 38√|tC2||tE2|(2 sin(2θα)− 3 sin(4θα)) cos(δE2 − δC2)
+ 1 332 70 |tC2||tE3|(278 sin(θα)− 455 sin(3θα)− 525 sin(5θα)) cos(δE3 − δC2)√
+ 218 |tC3||tE1|(2 sin(2θα)− 5 sin(4θα)) cos(δE1 − δC3)√
+ 1051√6 |tC3||tE2|(2 sin(θα)− 3 sin(3θα)− 5 sin(5θα)) cos(δE2 − δC3) }
− 7 364 2 |tC3||tE3|(13 sin(2θα) + 20 sin(4θα) + 25 sin(6θα)) cos(δE3 − δC3) · cos(φα),
RTT =−RT cos(2φα).
Using the Rodrigues’ formula Eq. (3.50) the Legendre polynomials up to n = 6
are
P0(x) = 1,
P1(x) = x,( )
P2(x) =
1 3x22( − 1 ,)
P3(x) =
1 3
2(5x − 3x ,1 )
P (x) = 35x4 − 30x24 8( + 3 , )
P5(x) =
1
8 (63x
5 − 70x3 + 15x , )
P6(x) =
1
16 231x
6 − 315x4 + 105x2 − 5 .
Using Eq. (3.51) the associated Legendre polynomials with m = 1 up to n = 6
are
√
P 11 (x) = − 1− x2,√
P 12 (x) = −3x( 1− x2),√
P 13 (
3
x) = − 2 22 5x − 1 1− x ,
133
Appendix A - Response Functions
5( )√
P 14 (x) = −2 7x
3 − 3x 1− x2,
1( ) = −15
( )
P x 21x4 − 14x2
√
5 8 ( + 1 1− x
2
) ,√
P 16 (x) = −
21 33x5 − 30x3 + 5x 1− x28 .
For x = cos(θ) the Legendre polynomials are given as
P0(cos(θ)) = 1,
P1(cos(θ)) = co(s(θ), )
P2(cos(θ)) =
1 3 cos22( (θ)− 1 , )
P3(cos(
1
θ)) = 5 cos32( (θ)− 3 cos(θ) , )
P4(cos(
1
θ)) = 8(35 cos
4(θ)− 30 cos2(θ) + 3 ,
1 )
P 55(cos(θ)) = 8 (63 cos (θ)− 70 cos
3(θ) + 15 cos(θ) , )
P6(cos(θ)) =
1
16 231 cos
6(θ)− 315 cos4(θ) + 105 cos2(θ)− 5 .
The associated Legendre polynomials for x = cos(θ) with 0 ≤ θ ≤ π in terms of
sine and cosine can be written as
P 11 (cos(θ)) = − sin(θ),
P 12 (cos(θ)) = −3 sin(θ) c(os(θ), )
P 13 (cos(θ)) = −
3
2 sin(θ) 5 cos
2
((θ)− 1 , )
P 14 (cos(
5
θ)) = −2 sin(θ) c(os(θ) 7 cos
2(θ)− 3 , )
P 15 (cos(θ)) = −
15
8 sin(θ) 21 cos
4(θ)− 14 cos2(θ) + 1 ,
1(cos( )) = −21
( )
P6 θ 8 sin(θ) cos(θ) 33 cos
4(θ)− 30 cos2(θ) + 5 .
134
APPENDIX B
Selected Data
In this chapter all nuclear data used for the calculations or the simulations is
given. Tables B.1 to B.4 represents the list of relevant adopted levels provided
by [NuDat]. In these tables, the errors are indicated separately by spaces. For
example, 5.57 eV 25 corresponds to 5.57(25) eV ≡ 5.57± 0.25 eV.
Table B.1: List of adopted levels for 8Be [NuDat].
E(level) Jπ Γ(level) E(γ) M(γ) Final level
(keV) (keV) Jπ(keV)
0 0+ 5.57 eV 25
α = 100
3030 10 2+ 1513 keV 15 3029 E2 0+(0)
α ≈ 100
11.35E+3 15 4+ ≈3.5 MeV 11350 E4 0+(0)
α ≈ 100
Table B.2: List of adopted levels for 12C [NuDat].
E(level) Jπ Γ(level) E(γ) I(γ) M(γ) Final level
(keV) (keV) Jπ(keV)
0 0+ STABLE
4439.82 21 2+ 10.8·10-3 eV 6 4438.94 100 E2 0+(0)
IT=100
7654.07 19 0+ 9.3 eV 9 3213.79 100 E2 2+(4439.82)
IT=4.16E-2
α ≈ 100
9641 5 3− 46 keV 3 9637 100 E3 0+(0)
α ≈ 100
IT < 4.1E-5
9870 60 2+ 850 keV 85
IT≈7.1E-6
α ≈ 100
135
Appendix B - Selected Data
Table B.3: List of adopted levels for 16O [NuDat].
E(level) Jπ Γ(level) E(γ) I(γ) M(γ) Final level
[keV] [keV] Jπ([keV])
0 0+ STABLE
6049.4 10 0+ 67 ps 5 6048.21 [E0] 0+(0)
6129.89 4 3− 18.4 ps 5 6128.634 100 [E3] 0+(0)
6917.1 6 2+ 4.70 fs 13 787.26 ≤0.008 [E1] 3−(6129.89)
867.712 0.027 3 [E2] 0+(6049.4)
6915.56 100 [E2] 0+(0)
7116.85 14 1− 8.3 fs 5 986.93 15 0.07014 [E2] 3−(6129.89)
1067.5 10 <6E-4 [E1] 0+(6049.4)
7115.15 14 100 [E1] 0+(0)
8871.9 5 2− 125 fs 11 1754.9 6 14.77 [M1+E2] 1−(7116.85)
1954.7 8 4.6 7 [E1] 2+(6917.1)
2741.5 5 100 21 [M1+E2] 3−(6129.89)
2822.2 12 0.15 5 [M2] 0+(6049.4)
8869.3 5 9.3 10 [M2] 0+(0)
9585 11 1− 420 keV 20 2688 11 12 4 [E1] 2+(6917.1)
IT = 6.7E-6 10 9582 11 100 16 [E1] 0+(0)
α = 100
9844.5 5 2+ 0.62 keV 10 2927.1 8 34 7 [M1] 2+(6917.1)
IT = 0.0016 3 3794.6 12 30 7 [E2] 0+(6049.4)
α = 100 9841.2 5 100 7 [E2] 0+(0)
10356 3 4+ 26 keV 3 3439 3 100 10 [E2] 2+(6917.1)
IT = 2.4E-4 4 4225 3 <1.6 [E1] 3−(6129.89)
α = 100 10352 3 9E-5 3 [E4] 0+(0)
10957 1 0− 5.5 fs 35 3839.6 10 100 [M1] 1−(7116.85)
11080 3 3+ < 12 keV
11096.7 16 4+ 0.28 keV 5 4179.0 17 81 20 [E2] 2+(6917.1)
IT = 0.0020 6 4966.0 16 100 42 [E1] 3−(6129.89)
α = 100
11260? (0+) 2500 keV
α = 100
11520 4 2+ 71 keV 3 4402 4 ≤0.9 [E1] 1−(7116.85)
IT = 9.4E-5 3 4602 4 4.4 11 [M1] 2+(6917.1)
α = 100 5470 5 4.6 8 [E2] 0+(6049.4)
11516 4 100.0 13 [E2] 0+(0)
11600 20 3− 800 keV 100
α = 100
12049 2 0+ 1.5 keV 5 12044.1 20 [E0] 0+(0)
IT = ?
α = 100
12440 2 1− 91 keV 6 6389.2 23 1.2 4 [E1] 0+(6049.4)
IT = 0.0132 24 12434.8 20 100 [E1] 0+(0)
p = 0.9 1
α = 99.1 1
12530 1 2− 0.111 keV 10 3657.7 12 67 4 [M1] 2−(8871.9)
IT = 3.2 3 5412.1 10 24.5 14 [M1] 1−(7116.85)
p = 14 7 5611.8 12 <2 [E1] 2+(6917.1)
α = 83 3 6398.7 10 100 4 [M1] 3−(6129.89)
12524.7 10 12.2 12 [M2] 0+(0)
12796 4 0− 40 keV 4 5678 4 100 [M1] 1−(7116.85)
IT = 0.0062 8
p = 100
12948.6 4 2− 1.34 keV 4 4096.1 7 84 4 [M1] 2−(8871.9)
IT = 0.28 3 5850.7 5 12 2 [M1] 1−(7116.85)
p = 78 4 6837.1 4 100 4 [M1] 3−(6129.89)
α = 22 4 12963.0 4 4.2 8 [M2] 0+(0)
13020 10 2+ 150 keV 10 13014 10 100 0+(0)
IT = ?
p = ?
α =?
136
Appendix B - Selected Data
13090 8 1− 130 keV 5 5972 8 3.1 8 [M1] 1−(7116.85)
IT = 0.026 4 7039 8 0.58 12 [E1] 0+(6049.4)
p = 71 13084 8 100 [E1] 0+(0)
α = 29
13129 10 3− 110 keV 30
IT = ?
p = 1
α = 99
13259 2 3− 21 keV 1
IT = ?
p = ?
α =?
13664 3 1+ 64 keV 3
p = 14
α = 86
IT < 0.0015
13869 2 4+ 89 keV 2
IT = ?
p = 0.6
α = 99.4
Table B.4: List of adopted levels for 20Ne [NuDat].
E(level) Jπ Γ(level) E(γ) I(γ) M(γ) Final level
(keV) (keV) Jπ(keV)
0 0+ STABLE
1633.674 15 2+ 0.73 ps 4 1633.602 15 100 [E2] 0+(0)
4247.7 11 4+ 64 fs 6 2613.8 11 100 [E2] 2+(1633.674)
4966.51 20 2− 3.3 ps 4 3332.54 20 99.4 2 [E1+M2+E3] 2+(1633.674)
4965.85 20 0.6 2 [M2] 0+(0)
5621.4 17 3− 139 fs 35 654.9 18 4.8 16 [M1] 2−(4966.51)
IT = 7 3 3987.3 17 87.6 10 [E1] 2+(1633.674)
α = 93 3 5620.6 17 7.6 10 [E3] 0+(0)
5787.7 26 1− 0.028 keV 3 4154 3 82 5 [E1] 2+(1633.674)
IT = 0.016 3 5787 3 18 5 [E1] 0+(0)
α = 100
6706 47
6725 5 0+ 19.0 keV 9 5090 5 100 [E2] 2+(1633.674)
IT = 1.7E-4 6724 5 [E0] 0+(0)
α = 100
7004.0 36 4− 305 fs 62 1383 4 25 [M1] 3−(5621.4)
2037 4 11 [E2] 2−(4966.51)
2756 4 63.5 [E1] 4+(4247.7)
5369 4 0.5 2 [M2] 2+(1633.674)
137
Appendix B - Selected Data
Table B.5: The atomic masses are from [Wan+17]. The nuclear masses marked
with ∗ are from [Hua+17]. The other nuclear masses are calculated with the formula
MN (A,Z) = MA(A,Z) − Z · me + Be(Z), with electron mass me and total electron
binding energy Be(Z). For Z = 1 th[e binding energy is Be(Z) = 13.6 eV a]nd for Z ≥ 2
the approximated formula Be(Z) = 14.4381 · Z2.39 + 1.55468 · 10−6 · Z5.35 eV was used.
This formula provides an rms error of 150 eV over the range up to Z = 106 [LPT03,
Appendix A] for the listed binding energies in [Hua+76]. In comparison to the listed
binding energies we assumed an error for Be of 2 eV for Z ≤ 4, 20 eV for Z = 6 and,
40 eV for Z ≥ 7, to calculate the error of the nuclear masses.
N Z A Element Atomic Mass Nuclear Mass Nuclear Mass
µu µu MeV/c2
0 1 1 H 1 007 825.032 241(94) 1 007 276.466 93(9)∗ 938.272 081 38(8)
1 1 1 D 2 014 101.778 114(122) 2 013 553.213 704(247) 1875.612 929(11)
1 2 4 He 3 016 029.322 645(220) 3 014 932.244 071(307) 2808.391 583(17)
2 2 4 He 4 002 603.254 130(63) 4 001 506.179 127(60)∗ 3727.379 378(23)
4 4 8 Be 8 005 305.10(4) 8 003 111.206(40) 7454.850 833(59)
6 6 12 C 12 000 000 11 996 709.642(21) 11 174.864 196(71)
7 7 14 N 14 003 074.004 460(207) 13 999 235.567(21) 13 040.205 271(82)
8 8 16 O 15 994 914.619 598(173) 15 990 528.212(43) 14 895.082 612(100)
10 10 20 Ne 19 992 440.1762(17) 19 986 958.182(43) 18 617.733 531(121)
12 12 24 Mg 23 985 041.697(14) 23 978 464.621(45) 22 335.798 211(143)
Table B.6: Calculated Gamow peaks for the reactions of the helium burning stage. Due
to the high density and temperature within the core of a star a plasma is formed. The
energy kT of around 10 – 20 keV is significantly higher than the binding energy of the
electrons so that the plasma consists of ions and free electrons. Thus, in the calculation
of the Gamow peak, nuclear masses are used instead of atomic masses. The Coulomb
penetration factor for each reaction at the Gamow peak is calculated using Eq. (2.12).
Reaction T EG ∆EG Coulomb penetration
GK keV keV factor
4He(α, γ)8Be 0.12 94.292 72.113 1.202× 10−8
4He(α, γ)8Be 0.2 132.549 110.380 2.088× 10−7
8Be(α, γ)12C 0.2 231.585 145.901 2.131× 10−12
12C(α, γ)16O 0.2 315.595 170.320 1.244× 10−16
16O(α, γ)20Ne 0.2 390.626 189.488 2.057× 10−20
20Ne(α, γ)24Mg 0.2 459.495 205.514 6.958× 10−24
138
Appendix B - Selected Data
Table B.7: Calculated Coulomb penetration factors Eq. (2.12) of the reactions of the
helium burning stage for a fixed cms energy Ecms = 300 keV. Due to the high density
and temperature within the core of a star a plasma is formed. The energy kT of around
10 – 20 keV is significantly higher than the binding energy of the electrons so that the
plasma consists of ions and free electrons. Thus, in the calculations, nuclear masses are
used instead of atomic masses.
Reaction Ecms Coulomb factor
keV
4He(α, γ)8Be 300 3.628× 10−5
8Be(α, γ)12C 300 5.565× 10−11
12C(α, γ)16O 300 4.859× 10−17
16O(α, γ)20Ne 300 3.433× 10−23
20Ne(α, γ)24Mg 300 2.189× 10−29
Table B.8: Physical constants used in this work. The quantities marked with ∗ are
from [Hua+17], the others are from [Zyl+20].
Quantity Symbol (equation) Value
speed of light in vacuum c 299 792 458m s−1
reduced Planck constant ~ 6.582 119 569× 10−22 MeVs
electron charge magnitude e 1.602 176 634× 10−19 C
Boltzmann constant k 1.380 649× 10−23 JK−1
conversion constant ~c 197.326 980 4MeV fm
atomic mass unit∗ u=(mass 12C atom)/12 931.494 095 4(57)MeV/c2
electron mass∗ me 548 579.909 070(16) nu
=0.510 998 946 1(31)MeV/c2
proton mass∗ Mp 1 007 276 466.93(9) nu
=938.272 081 38(8)MeV/c2
alpha particle mass∗ Mα 4 001 506 179.127(60) nu
=3727.379 378(23)MeV/c2
fine-structure constant αem = e2/4πε0~c 7.297 352 569 3(11)× 10−3
=1/137.035 999 084(21)
139
Appendix B - Selected Data
Table B.9: Coordinate system for the target and generator system. The angles are
defined analogously to the spherical coordinate system.
Coordinate Definition
x up
y right with the beam direction y = z × x
z beam direction
Origin Target middle point (0, 0, 0)
θ Angle from the beamline within in the range [0, π)
φ Rotation angle in x− y plane in range [−π, π)
Table B.10: Base units used in the simulation software
Quantity Unit
Length mm
Cross Section µb = 10−30 cm2
Angle rad
Energy MeV
Momentum MeV c−1
Mass MeV c−2
Charge e
Beam Current µA
Time ns
Temperature K
Table B.11: To extrapolate the S-factor different step length are used for different
energy ranges.
Ecms cmsmin [MeV] Emax[MeV] dE[MeV]
0.01 6 0.001
5.5 14 0.005
12 20 0.01
140
APPENDIX C
EPICS Support Files
In this chapter the installation scripts of EPICS, MXSlowControl and the modules
are presented. In addition different EPICS related C++ header and source files are
displayed.
Code C.1: Installation Script EPICS
File: setup-epics-ioc-raspbian.sh
1 #!/bin/bash 29 #wget https://github.com/epics-modules/ c
2 # author: Stefan Lunkenheimer ↪→ autosave/archive/ c
↪→ (stelunke@uni-mainz.de) ↪→ R${autosaveVersion}.tar.gz -O
3 # Basic file on: https://gist.github.com/ c ↪→ autosave-R${autosaveVersion}.tar.gz ||
↪→ inigoalonso/a77314206b516e94d4aa ↪→ exit
4 30
5 epicsDir=/opt/epics 31
6 gitDir="$(cd "$( dirname "$0")/.." && pwd )" 32 # Preliminiary installations!!!
7 epicsVersion=3.14.12.6 33 sudo apt-get install libreadline7
8 asynVersion=asyn4-31 ↪→ libreadline-dev
9 streamVersion=StreamDevice-2 34
10 streamSubVersion=6 35 # ------------------------------------------
11 autosaveVersion=5-7-1 36 # ----------- INSTALL EPICS BASE -----------
12 DOWNLOADS=${gitDir}/install/external 37 # ------------------------------------------
13 38
14 pyepics=true 39 # Create a place to work
15 frontend=true 40 echo "Creating a place to work: ${epicsDir}"
16 41 sudo mkdir -p ${epicsDir}
17 export LC_ALL=en_US.UTF-8 42 sudo chown ${USER}:${GROUP} ${epicsDir}
18 export LC_CTYPE=en_US.UTF-8 43
19 export LANG=en_US.UTF-8 44 echo "Untar-ing EPICS base"
20 45 tar -zxf
21 ↪→ ${DOWNLOADS}/baseR${epicsVersion}.tar.gz
22 # Download the necessary source files ↪→ -C ${epicsDir}/ || exit
23 echo "Downloading the necessary source files" 46
24 47 echo "Creating links to epics base folder"
25 cd ${DOWNLOADS} 48 ln -s ${epicsDir}/base-${epicsVersion}
26 #wget http://www.aps.anl.gov/epics/download/ c ↪→ ${epicsDir}/base
↪→ base/baseR${epicsVersion}.tar.gz || 49
↪→ exit 50 # Set the EPICS environment variables --
27 #wget http://www.aps.anl.gov/epics/download/ c ↪→ Create thisEPICS.sh
↪→ modules/${asynVersion}.tar.gz || 51 echo "Setting up the EPICS enviroment
↪→ exit ↪→ variables"
28 #wget http://epics.web.psi.ch/software/ c 52 cat << EOF > ${epicsDir}/thisEPICS.sh
↪→ streamdevice/${streamVersion}.tgz || 53 #EPICS
↪→ exit 54 export EPICS_ROOT=${epicsDir}
55 export EPICS_BASE=\${EPICS_ROOT}/base
56 export EPICS_HOST_ARCH=\$(\${EPICS_BASE}/ c
↪→ startup/EpicsHostArch)
141
Appendix C - EPICS Support Files
57 export EPICS_BASE_BIN=\${EPICS_BASE}/bin/ c 120 echo "ASYN=${EPICS_ROOT}/modules/asyn" >>
↪→ \${EPICS_HOST_ARCH} ↪→ ${EPICS_ROOT}/RELEASE.local
58 export EPICS_BASE_LIB=\${EPICS_BASE}/lib/ c 121 # --------------------------------------
↪→ \${EPICS_HOST_ARCH} 122
59 if [ "" = "\${LD_LIBRARY_PATH}" ]; then 123 # -------- Install StreamDevice --------
60 export LD_LIBRARY_PATH=\${EPICS_BASE_LIB} 124 echo "Installing StreamDevice library into
61 else ↪→ EPICS"
62 export LD_LIBRARY_PATH= c 125 mkdir -p ${EPICS_ROOT}/modules/stream
→ \${EPICS_BASE_LIB}:\${LD_LIBRARY_PATH} 126 cd ${EPICS_ROOT}/modules/stream↪
63 fi 127 echo "Untar-ing"
64 export PATH=\${PATH}:\${EPICS_BASE_BIN} 128 tar -zxvf ${DOWNLOADS}/${streamVersion}.tgz
65 ↪→ -C ${EPICS_ROOT}/modules/stream || exit
66 export HOST=\$(hostname) 129
67 EOF 130 echo "Configuring"
68 131 echo | makeBaseApp.pl -t support
69 chmod +x ${epicsDir}/thisEPICS.sh 132 echo "ASYN=${EPICS_ROOT}/modules/asyn" >>
70 ↪→ configure/RELEASE.local
71 # Set envoirement Variables for the 133 echo "Compiling 1"
↪→ installation 134 make -j4 || exit
72 source ${epicsDir}/thisEPICS.sh 135
73 136 cd ${streamVersion}-${streamSubVersion} ||
74 # Configure RELEASE ↪→ exit
75 echo "EPICS_BASE=${EPICS_ROOT}/base" > 137 #
↪→ ${EPICS_ROOT}/RELEASE.local 138 # ------ Insert PROFIBUS BUS API to
76 ↪→ StreamDevice ------
77 # Compile EPICS 139 echo "Copy Profibus src-files"
78 echo "Compiling EPICS" 140 cp ${gitDir}/install/src/* src/ || exit
79 cd ${EPICS_BASE} 141 echo "Add Profibus to config_stream"
80 make -j4 || exit 142 cat << EOF >> src/CONFIG_STREAM
81 143
82 # --------------------------------------- 144 #PROFIBUS
83 # ----------- INSTALL MODULES ----------- 145 BUSSES += Profibus
84 # --------------------------------------- 146 CPPFLAGS += -std=c++11
85 147 CXXFLAGS += -std=c++11
86 # ------------- Install ASYN ------------- 148 EOF
87 echo "Installing ASYN library into EPICS" 149 echo "Compiling 2"
88 mkdir -p ${EPICS_ROOT}/modules 150 make -j4 || exit
89 echo "Untar-ing" 151
90 tar -zxf ${DOWNLOADS}/${asynVersion}.tar.gz 152 # Modify thisEPICS.sh
↪→ -C ${EPICS_ROOT}/modules/ || exit 153 echo "export
91 echo "Linking" ↪→ STREAM=\${EPICS_ROOT}/modules/stream" >>
92 ln -s ${EPICS_ROOT}/modules/${asynVersion} ↪→ ${EPICS_ROOT}/thisEPICS.sh
→ ${EPICS_ROOT}/modules/asyn 154 export STREAM=${EPICS_ROOT}/modules/stream↪
93 155
94 # MODIFY RELEASE 156 # Modify RELEASE.local
95 echo "Editing the configuration" 157 echo "STREAM=${EPICS_ROOT}/modules/stream" >>
96 cd ${EPICS_ROOT}/modules/asyn ↪→ ${EPICS_ROOT}/RELEASE.local
97 sed -ie '/IPAC/ s/^#*/#/' configure/RELEASE 158 # ------------------------------------
98 sed -ie '/SNCSEQ/ s/^#*/#/' configure/RELEASE 159
99 sed -ie '/EPICS_BASE/ s/^#*/#/' 160 # --------- Install AUTOSAVE ---------
→ configure/RELEASE 161 echo "Installing Autosave Module into EPICS"↪
100 cat << EOF >> configure/RELEASE 162 echo "Untar-ing"
101 163 tar -zxvf ${DOWNLOADS}/autosave- c
102 # These allow developers to override the ↪→ R${autosaveVersion}.tar.gz -C
↪→ RELEASE variable settings ↪→ ${EPICS_ROOT}/modules/ || exit
103 # without having to modify the 164 echo "Linking"
→ configure/RELEASE file itself. 165 ln -s ${EPICS_ROOT}/modules/autosave-↪ c
104 #-include \$(TOP)/../RELEASE.local ↪→ R${autosaveVersion}
105 #-include \$(TOP)/configure/RELEASE.local ↪→ ${EPICS_ROOT}/modules/autosave
106 EPICS_BASE=${EPICS_BASE} 166 cd ${EPICS_ROOT}/modules/autosave
107 EOF 167
108 168 # MODIFY RELEASE
109 # Compile ASYN 169 echo "Editing the configuration"
110 echo "Compiling" 170 sed -ie '/EPICS_BASE/ s/^#*/#/'
111 cd ${EPICS_ROOT}/modules/asyn ↪→ configure/RELEASE
112 make -j4 || exit 171 cat << EOF >> configure/RELEASE
113 172 # These allow developers to override the
114 # Modify thisEPICS.sh ↪→ RELEASE variable settings
115 echo -e "\n#Modules" >> 173 # without having to modify the
↪→ ${EPICS_ROOT}/thisEPICS.sh ↪→ configure/RELEASE file itself.
116 echo "export 174 #-include \$(TOP)/../RELEASE.local
→ ASYN=\${EPICS_ROOT}/modules/asyn" >> 175 #-include \$(TOP)/configure/RELEASE.local↪
↪→ ${EPICS_ROOT}/thisEPICS.sh 176 EPICS_BASE=${EPICS_ROOT}/base
117 export ASYN=${EPICS_ROOT}/modules/asyn 177 EOF
118 178
119 # Modify RELEASE.local 179 echo "Compiling"
142
Appendix C - EPICS Support Files
180 cd ${EPICS_ROOT}/modules/autosave 211 export
181 make -j4 || exit ↪→ DEVGPIO=${EPICS_ROOT}/modules/devgpio
182 212
183 # Modify thisEPICS.sh 213 # Modify RELEASE.local
184 echo "export 214 echo "DEVGPIO=${EPICS_ROOT}/modules/ c
↪→ AUTOSAVE=\${EPICS_ROOT}/modules/autosave" ↪→ devgpio" >>
↪→ >> ${EPICS_ROOT}/thisEPICS.sh ↪→ ${EPICS_ROOT}/RELEASE.local
185 export 215 fi
↪→ AUTOSAVE=${EPICS_ROOT}/modules/autosave 216 # -------------------------------------
186 217
187 # Modify RELEASE.local 218 # ---------- Install pyEPICS ----------
188 echo 219 if [ $pyepics = true ] ;
↪→ "AUTOSAVE=${EPICS_ROOT}/modules/autosave" 220 then
↪→ >> ${EPICS_ROOT}/RELEASE.local 221 sudo apt-get install python-pip ||
189 # -------------------------------------- 222 ( echo -e
190 ↪→ "\n\n------------------------ c
191 # ---------- Install dev-gpio ---------- ↪→ \nInstalling python-pip failed"
192 # Testing on RaspberyPi ↪→ && exit )
193 if [ $EPICS_HOST_ARCH = "linux-arm" ] 223 sudo pip install pyepics ||
194 then 224 ( echo -e
195 gpioVersion=master ↪→ "\n\n------------------------ c
196 cd ${DOWNLOADS} ↪→ \nInstalling pyepics failed" &&
197 #wget -O dev-gpio_${gpioVersion}.zip ↪→ exit )
↪→ https://github.com/ffeldbauer/epics- c 225
↪→ devgpio/archive/${gpioVersion}.zip || 226 # Modify thisEPICS.sh
↪→ exit 227 echo -e "\n#PYEPICS" >>
198 unzip ${DOWNLOADS}/dev- c ↪→ ${EPICS_ROOT}/thisEPICS.sh
↪→ gpio_${gpioVersion}.zip -d 228 echo "export PYEPICS_LIBCA= c
↪→ ${EPICS_ROOT}/modules || exit ↪→ \${EPICS_BASE_LIB}/libca.so" >>
199 ln -s ${EPICS_ROOT}/modules/epics- c ↪→ ${EPICS_ROOT}/thisEPICS.sh
↪→ devgpio-${gpioVersion} 229 fi
↪→ ${EPICS_ROOT}/modules/devgpio || exit 230 # ---------------------------------------
200 cd ${EPICS_ROOT}/modules/devgpio 231
201 232 # ----------- Frontend Module -----------
202 # MODIFY RELEASE 233 if [ $frontend = true ] ;
203 echo "Editing the configuration" 234 then
204 sed -ie '/EPICS_BASE/ s/^#*/#/' 235 $gitDir/modules/frontend/ c
↪→ configure/RELEASE ↪→ install_module_frontend.sh
205 echo "EPICS_BASE=${EPICS_BASE}" >> 236 fi
↪→ configure/RELEASE 237 # ---------------------------------------
206 238
207 make -j4 || exit 239 # --------- Install Environment ---------
208 240 sudo install ${EPICS_ROOT}/thisEPICS.sh
209 # Modify thisEPICS.sh ↪→ /etc/profile.d/epics.sh
210 echo "export DEVGPIO=\${EPICS_ROOT}/ c 241
↪→ modules/devgpio" >> 242 echo "Done!"
↪→ ${EPICS_ROOT}/thisEPICS.sh
Code C.2: Installation Script MXSlowControl
File: install_MXSlowControl.sh
1 #!/bin/bash 20 echo | ${EPICS_BASE}/bin/${EPICS_HOST_ARCH}/ c
2 # author: Stefan Lunkenheimer ↪→ makeBaseApp.pl -t ioc -i
↪→ (stelunke@uni-mainz.de) ↪→ ${iocNAME}
3 21
4 # Remark: Check if the export variables are 22 cp ${gitDir}/MXSlowControl/CONFIG_SITE.local
↪→ set correctly! ↪→ ${EPICS_ROOT}/ || exit
5 source /etc/profile.d/epics.sh 23
6 24 echo "Add modules to Base Application"
7 iocNAME="MXSlowControl" 25 # Add streamDevice and Asyn to the ioc
8 26 sed -e "/${iocNAME}_DBD += xxx.dbd/ a
9 gitDir="$(cd "$( dirname "$0")/.." && pwd )" ↪→ ${iocNAME}_DBD += stream.dbd asyn.dbd
10 ↪→ drvAsynSerialPort.dbd drvAsynIPPort.dbd
11 echo "Install $iocName" ↪→ asSupport.dbd" \
12 27 -e "/${iocNAME}_LIBS += xxx/ a
13 cd ${EPICS_ROOT} ↪→ ${iocNAME}_LIBS += stream asyn
14 echo "Generate Base Application" ↪→ autosave" \
15 mkdir -p ${iocNAME} || exit 28 -i ${iocNAME}App/src/Makefile
16 cd ${iocNAME} 29
17 $ASYN/bin/$EPICS_HOST_ARCH/makeSupport.pl -t 30 # Add dev-gpio to the ioc
↪→ streamSCPI ${iocNAME} || exit 31 if [ $EPICS_HOST_ARCH = "linux-arm" ]
18 rm -rf configure 32 then
19 $EPICS_BASE/bin/$EPICS_HOST_ARCH/ c 33 sed -e "/${iocNAME}_DBD += xxx.dbd/ a
↪→ makeBaseApp.pl -t ioc ↪→ ${iocNAME}_DBD += devgpio.dbd" \
↪→ ${iocNAME}
143
Appendix C - EPICS Support Files
34 -e "/${iocNAME}_LIBS += xxx/ a 76 echo "Transfer Basic db and st.cmd"
↪→ ${iocNAME}_LIBS += devgpio" \ 77 rm -f db/*
35 -i ${iocNAME}App/src/Makefile 78 cp -r ${gitDir}/MXSlowControl/db/* db/ ||
36 fi ↪→ exit
37 79 cp -r ${gitDir}/MXSlowControl/iocBoot/ c
38 # Add Device Support components ↪→ iocMXSlowControl/* iocBoot/ioc${iocNAME}/
39 echo "Add Device Support Drivers" ↪→ || exit
40 # Install necessary librarys 80 sed -e s/EPICS_HOST_ARCH/${EPICS_HOST_ARCH}/
41 sudo apt-get install libusb-dev ↪→ \
42 cp -r ${gitDir}/MXSlowControl/App/src/* 81 -e s/YOUR_HOSTNAME/$(hostname)/ \
↪→ ${iocNAME}App/src 82 -i iocBoot/ioc${iocNAME}/st.cmd
43 83 chmod +x iocBoot/ioc${iocNAME}/st.cmd
44 # Add USBADC and USBDAC to the ioc Makefile 84
45 sed -e "7i LIBRARY_IOC += dbLoadMultiRecords c 85 echo "Transfer autosave example"
↪→ \ndbLoadMultiRecords_SRCS += 86 mkdir -p autosave || exit
↪→ dbLoadMultiRecords.cpp c 87 cp -r ${gitDir}/MXSlowControl/autosave/*
↪→ \ndbLoadMultiRecords_LIBS += ↪→ autosave/ || exit
↪→ \$(EPICS_BASE_IOC_LIBS)\n" \ 88
46 -e "/${iocNAME}_DBD += xxx.dbd/ a 89 echo "Transfer udev/rules"
↪→ ${iocNAME}_DBD += 90 sudo cp ${gitDir}/MXSlowControl/rules/999- c
↪→ dbLoadMultiRecords.dbd" \ ↪→ mxslowcontrol.rules /etc/udev/rules.d/ ||
47 -e "/${iocNAME}_LIBS += xxx/ a ↪→ exit
↪→ ${iocNAME}_LIBS += 91
↪→ dbLoadMultiRecords" \ 92 # INSTALLING service
48 -i ${iocNAME}App/src/Makefile 93 sudo apt-get install tmux
49 94 sudo install ${gitDir}/MXSlowControl/ c
50 # Add USBADC and USBDAC to the ioc Makefile ↪→ systemctl/epics.service
51 sed -e "7i LIBRARY_IOC += ↪→ /etc/systemd/system/epics.service
↪→ aoUSBDAC\naoUSBDAC_SRCS += devUSBDAC.c 95 sudo sed -i 's/\$IOC/'$(echo
↪→ USBDAC.cpp\naoUSBDAC_LIBS += ↪→ $EPICS_ROOT/MXSlowControl/iocBoot/ c
↪→ \$(EPICS_BASE_IOC_LIBS)\n" \ ↪→ iocMXSlowControl|sed 's/\//\\\//g')'/'
52 -e "7i LIBRARY_IOC += ↪→ /etc/systemd/system/epics.service
↪→ aiUSBADC\naiUSBADC_SRCS += 96 sudo install ${gitDir}/MXSlowControl/ c
↪→ devUSBADC.c USBADC.cpp\naiUSBADC_LIBS ↪→ systemctl/caArchiver.service
↪→ += \$(EPICS_BASE_IOC_LIBS)\n" \ ↪→ /etc/systemd/system/caArchiver.service
53 -e "7i USR_CXXFLAGS += -std=c++11\n" \ 97 sudo systemctl daemon-reload
54 -e "/${iocNAME}_DBD += xxx.dbd/ a 98
↪→ ${iocNAME}_DBD += aiUSBADC.dbd 99 # update environments
↪→ aoUSBDAC.dbd" \ 100 cat << EOF >> ${EPICS_ROOT}/thisEPICS.sh
55 -e "/${iocNAME}_LIBS += xxx/ a 101
↪→ ${iocNAME}_LIBS += aiUSBADC 102 #MXSlowControl
↪→ aoUSBDAC\n${iocNAME}_SYS_LIBS += usb" 103 export MXSLOWCONTROL=\${EPICS_ROOT}/ c
↪→ \ ↪→ MXSlowControl
56 -i ${iocNAME}App/src/Makefile 104 export MXSLOWCONTROL_BIN=\${MXSLOWCONTROL}/ c
57 ↪→ bin/\${EPICS_HOST_ARCH}
58 # add caArchiver 105 export MXSLOWCONTROL_LIB=\${MXSLOWCONTROL}/ c
59 sed -i "7i PROD_HOST +=
↪→ lib/\${EPICS_HOST_ARCH}
↪→ caArchiver\ncaArchiver_SRCS += 106 export IOC=\${MXSLOWCONTROL}/iocBoot/
↪→ caArchiver.c\ncaArchiver_LIBS += c
\$(EPICS_BASE_HOST_LIBS)\n" ↪→ iocMXSlowControl↪→
iocNAME App/src/Makefile 107 if [ "" = "\${LD_LIBRARY_PATH}" ]; then↪→ ${ }
mkdir EPICS_ROOT /log 108 export60 ${ }
touch EPICS_ROOT /CA_ADDR_LIST.local ↪→ LD_LIBRARY_PATH=\${MXSLOWCONTROL_LIB}61 ${ }
62 touch ${EPICS_ROOT 109 else}/log/log.txt
110 export LD_LIBRARY_PATH=\${MXSLOWCONTROL
63 c
# module Frontend ↪→ _LIB}:\${LD_LIBRARY_PATH}64
111 fi
65 if [ "$(grep FRONTEND
$EPICS_ROOT/RELEASE.local)" != "" ] 112 export PATH=\${PATH}:\${MXSLOWCONTROL_BIN}↪→
66 then 113
sed -e "/${iocNAME}_DBD += xxx.dbd/ a 114 #Channel Access List67
${iocNAME}_DBD += frontend.dbd" \ 115 export EPICS_CA_ADDR_LIST="\$(<\${EPICS↪→ c
68 -e "/${iocNAME}_LIBS += xxx/ a ↪→ _ROOT}/CA_ADDR_LIST.local)"
→ ${iocNAME}_LIBS += frontend" \ 116↪
69 -i ${iocNAME}App/src/Makefile 117 #caArchiver
70 fi 118 export CA_ARCHIVER_LOG=\${EPICS_ROOT}/log
71 119 EOF
72 # BIULD ALL 120 sudo install ${EPICS_ROOT}/thisEPICS.sh
73 echo "Build Application" ↪→ /etc/profile.d/epics.sh
74 make -j4 || exit 121
75 122 echo "DONE!"
144
Appendix C - EPICS Support Files
Code C.3: Installation Script Module MXFrontend
File: install_module_frontend.sh
1 #!/bin/bash 24 cp $SOURCE_DIR/src/*.h ${moduleName}App/src/.
2 # author: Stefan Lunkenheimer ↪→ || (echo "${RED}Error while cp h" &&
↪→ (stelunke@uni-mainz.de) ↪→ exit)
3 25 cp $SOURCE_DIR/src/*.dbd
4 # Remark: Check if the export variables are ↪→ ${moduleName}App/src/. || (echo
↪→ set correctly! ↪→ "${RED}Error while cp dbd" && exit)
5 RED='\033[1;31m' 26
6 HINT='\033[1;36m' 27 # Configure Makefile
7 NC='\033[0m' # No Color 28 sed -e "/${moduleName}_LIBS +=/r
8 ↪→ $SOURCE_DIR/src/Makefile_ext" \
9 #source /etc/profile.d/epics.sh 29 -i ${moduleName}App/src/Makefile || (echo
10 moduleName="frontend" ↪→ "${RED}Error sed Makefile" && exit)
11 modulePath="FRONTEND" 30
12 31 # Make all
13 SOURCE_DIR="$(cd "$( dirname "$0")" && pwd 32 make -j4 || (echo "${RED}Error while make" &&
↪→ )" ↪→ exit)
14 33
15 cd $EPICS_ROOT/modules || exit 34 # Modify thisEPICS.sh
16 mkdir -p $moduleName || exit 35 echo "export $modulePath=${EPICS_ROOT}/ c
17 ↪→ modules/$moduleName" >>
18 # Make Base Application Support Module ↪→ ${EPICS_ROOT}/thisEPICS.sh
19 cd $moduleName 36 export $modulePath=${EPICS_ROOT}/modules/ c
20 echo $moduleName | makeBaseApp.pl -t support ↪→ $moduleName
↪→ || (echo "${RED}Error makeBaseApp" && 37
↪→ exit) 38 # Modify RELEASE.local
21 39 echo "$modulePath=${EPICS_ROOT}/modules/ c
22 # Copy source files ↪→ $moduleName" >>
23 cp $SOURCE_DIR/src/*.cpp ↪→ ${EPICS_ROOT}/RELEASE.local
↪→ ${moduleName}App/src/. || (echo 40
↪→ "${RED}Error while cp cpp" && exit) 41 # Done
42 echo "${HINT}\n\nmodule $moduleName
↪→ installed${NC}"
Code C.4: Bash completion for EPICS
File: epics_BASH_completion.sh
1 # auto completion for a running softIOC 20 else
2 # 21 opts=$(cat ${EPICS_ROOT}/dbl.txt)
3 # Tutorials / Sources: 22 fi
4 # https://web.archive.org/web/ c 23
↪→ 20200507173259/https://debian- c 24 if [[ ${cur} == *.* ]] ; then
↪→ administration.org/ 25 local cur1=${cur%.*}
↪→ article/317/ c 26 #local cur2=${cur##*.}
↪→ An_introduction_to_bash_completion_part_2 27 local param="NAME DESC SCAN PREC EGU
5 # https://www.gnu.org/savannah-checkouts/ c ↪→ INPA INPB CALC HOPR LOPR VAL"
↪→ gnu/bash/manual/bash.html 28 local fields=$(for x in $param; do
↪→ #A-Programmable-Completion-Example ↪→ echo ${cur1}.${x}; done)
6 # 29 COMPREPLY=($(compgen -W "${fields}"
7 # author: Stefan Lunkenheimer ↪→ -- "${cur}"))
↪→ (stelunke@uni-mainz.de) 30 return 0
8 # date: 2018-06-13 31 else
9 # Version 1.2 32 COMPREPLY=( $(compgen -W "${opts}"
10 ↪→ -- "${cur}"))
11 _epics() 33 return 0
12 { 34 fi
13 local cur prev opts 35
14 COMPREPLY=() 36 return 0
15 cur="${COMP_WORDS[COMP_CWORD]}" 37 }
16 prev="${COMP_WORDS[COMP_CWORD-1]}" 38 complete -F _epics caget
17 if [ -e ${EPICS_ROOT}/dbl_global.txt ] 39 complete -F _epics caput
18 then 40 complete -F _epics camonitor
19 opts=$(cat ${EPICS_ROOT}/dbl.txt 41 complete -F _epics logreader
↪→ ${EPICS_ROOT}/dbl_global.txt |
↪→ sort | uniq)
145
Appendix C - EPICS Support Files
Code C.5: ZSH completion for EPICS
File: epics_ZSH_completion.sh
1 # auto completion for a running softIOC 17 else
2 # 18 cmds=( ${(uf)"$(<${EPICS_ROOT}/dbl.txt)"}
3 # Tutorials / Sources: ↪→ )
4 # https://github.com/zsh-users/zsh- c 19 fi
↪→ completions/blob/master/zsh-completions- c 20
↪→ howto.org 21 # description: The character ':' makes
5 # https://stackoverflow.com/questions/ c ↪→ Problems, because the description uses it
↪→ 17318913/make-zsh-complete-arguments- c ↪→ as seperator for their individual
↪→ from-a-file ↪→ description.
6 # 22 # There are two Solutions:
7 # author: Stefan Lunkenheimer 23 # In the file use \: insteead of : then
↪→ (stelunke@uni-mainz.de) ↪→ everything works fine
8 # date: 2018-06-13 24 # use the argument function
9 # Version 1.2 25
10 26 #_describe 'a description of the completion
11 _epics() ↪→ options' cmds
12 { 27 _arguments '*:foo:(${cmds})'
13 cmds= 28 }
14 if [ -e ${EPICS_ROOT}/dbl_global.txt ] 29
15 then 30 compdef _epics caget caput camonitor
16 cmds=( ${(uf)"$(cat ${EPICS_ROOT}/dbl.txt ↪→ logreader
↪→ ${EPICS_ROOT}/dbl_global.txt | sort |
↪→ uniq)"} )
Code C.6: Header file of the C++ class for the MUX0 register of the ADS1248
chip.
File: MUX0.h
1 /* 34 };
2 * MUX0.h 35
3 * 36 class Register
4 * Created on: Oct 16, 2018 37 {
5 * Author: stelunke (stelunke@uni-mainz.de) 38 public:
6 */ 39 Register ( );
7 40
8 #ifndef MUX0_H_ 41 Register (BCS, AIN, AIN );
9 #define MUX0_H_ 42
10 43 void setBCS(BCS);
11 #include <cstdint> 44
12 45 void setMUX_SP(AIN);
13 namespace MUX0 46
14 { 47 void setMUX_SN(AIN);
15 48
16 enum class BCS : std::uint8_t 49 std::uint8_t getBitset();
17 { 50
18 off = 0b00, 51 private:
19 on_0_5uA = 0b01, 52 BCS m_bcs;
20 on_2uA = 0b10, 53 AIN m_muxSP;
21 on_10uA = 0b11 54 AIN m_muxSN;
22 }; 55
23 56 enum Bit
24 enum class AIN : std::uint8_t 57 {
25 { 58 BCS_BIT=6,
26 AIN0 = 0x0, 59 MUX_SP_BIT=3,
27 AIN1 = 0x1, 60 MUX_SN_BIT=0
28 AIN2 = 0x2, 61 };
29 AIN3 = 0x3, 62 };
30 AIN4 = 0x4, 63
31 AIN5 = 0x5, 64 } /* namespace MUX0 */
32 AIN6 = 0x6, 65
33 AIN7 = 0x7 66 #endif /* MUX0_H_ */
146
Appendix C - EPICS Support Files
Code C.7: Source file of the C++ class for the MUX0 register of the ADS1248
chip.
File: MUX0.cpp
1 /* 30 {
2 * MUX0.cpp 31 m_bcs = bcs;
3 * 32 }
4 * Created on: Oct 16, 2018 33
5 * Author: stelunke (stelunke@uni-mainz.de) 34 void Register::setMUX_SP(AIN muxSP)
6 */ 35 {
7 36 m_muxSP = muxSP;
8 #include "MUX0.h" 37 }
9 38
10 namespace MUX0 39 void Register::setMUX_SN(AIN muxSN)
11 { 40 {
12 41 m_muxSN = muxSN;
13 Register::Register ( ): 42 }
14 m_bcs{BCS::off}, 43
15 m_muxSP{AIN::AIN0}, 44 std::uint8_t Register::getBitset()
16 m_muxSN{AIN::AIN1} 45 {
17 { 46 std::uint8_t bitset = 0x00;
18 47 bitset |=
19 } ↪→ static_cast<std::uint8_t>(m_bcs) <<
20 ↪→ BCS_BIT;
21 Register::Register (BCS bcs, AIN muxSP, AIN 48 bitset |=
↪→ muxSN): ↪→ static_cast<std::uint8_t>(m_muxSP) <<
22 m_bcs{bcs}, ↪→ MUX_SP_BIT;
23 m_muxSP{muxSP}, 49 bitset |=
24 m_muxSN{muxSN} ↪→ static_cast<std::uint8_t>(m_muxSN) <<
25 { ↪→ MUX_SN_BIT;
26 50 return bitset;
27 } 51 }
28 52
29 void Register::setBCS(BCS bcs) 53 } /* namespace MUX0 */
Code C.8: Header file of the C++ class for the I 2C communication of the SiDet
device driver.
File: RPiI2C.h
1 /** 27 {
2 * \class RPiI2C 28 public:
3 * 29 /// Delete Copy constructor for Singleton
4 * \brief I2C connection class for Raspberry 30 RPiI2C(const RPiI2C&) = delete;
↪→ Pi 31
5 * 32 /// Delete Move constructor for Singleton
6 * Singleton to handle the communication for 33 RPiI2C(const RPiI2C&&) = delete;
↪→ the I2C Bus of a Raspberry Pi 34
↪→ (/dev/i2c-1). 35 /// Delete Copy assignment operator for
7 * ↪→ Singleton
8 * \todo For a RPi4 multiple I2C Bus 36 RPiI2C& operator= (const RPiI2C) =
↪→ connections are possible ↪→ delete;
9 * 37
10 * \author Stefan Lunkenheimer 38 /// Default deconstructor
11 * Contact: stelunke@uni-mainz.de 39 ~RPiI2C();
12 * 40
13 * Created on: Thu Feb 27th, 2020 41 /**
14 * 42 * \brief write data to I2C bus
15 * Source: https://raspberry- c 43 *
↪→ projects.com/pi/programming-in-c/i2c/ c 44 * write @ref buffer of length @ref len
↪→ using-the-i2c-interface ↪→ to a device on I2C bus with the adress
16 */ ↪→ @ref slaveAddr
17 45 * @param slaveAddr I2C Adress of the
18 #ifndef RPII2C_H ↪→ device
19 #define RPII2C_H 46 * @param buffer Data to write to device
20 47 * @param len Length of the data buffer
21 #include <cstdint> 48 * @return <br>true</br>: write to device
22 #include <string> ↪→ succesfull; <br>false</br>: write to
23 ↪→ device failed
24 #include "mxlog.h" 49 */
25 50 bool
26 class RPiI2C final
147
Appendix C - EPICS Support Files
51 writeI2C(uint8_t slaveAddr, uint8_t* 99 * @return <br>true</br> I2C connection
↪→ buffer, uint8_t len); ↪→ ready; <br>false</br> I2C connection
52 ↪→ failed
53 /** 100 */
54 * \brief read data from I2C bus 101 bool
55 * 102 open();
56 * read data @ref buffer of length @ref 103
↪→ len from device on I2C bus with the 104 /**
↪→ adress @ref slaveAddr 105 * \brief reopen I2C connection
57 * @param slaveAddr I2C Adress of the 106 *
↪→ device 107 * @return <br>true</br> I2C connection
58 * @param buffer Data from device ↪→ ready; <br>false</br> reopen failed
59 * @param len Length of the data buffer 108 */
60 * @return <br>true</br>: write to device 109 bool
↪→ succesfull; <br>false</br>: write to 110 reopen();
↪→ device failed 111
61 */ 112 /**
62 bool 113 * \brief close I2C connection
63 readI2C(uint8_t slaveAddr, uint8_t* 114 *
↪→ buffer, uint8_t len); 115 * @return <br>true</br> I2C closed;
64 ↪→ <br>false</br> failed
65 /** 116 */
66 * \brief status of the I2C connection 117 bool
67 * 118 close();
68 * @return <br>true</br> I2C connection 119
↪→ ready; <br>false</br> I2C connection 120 /**
↪→ closed 121 * \brief set log level
69 */ 122 *
70 bool 123 * @param new log level
71 getStatus() const; 124 */
72 125 void
73 /** 126 set_level(spdlog::level::level_enum
74 * \brief get the class Identifier ↪→ level);
75 * 127
76 * @return class name 128 /**
77 */ 129 * \brief set log level
78 std::string 130 *
79 IsA() const; 131 * @param new log level
80 132 */
81 /** 133 void
82 * \brief get Instance of the Singleton 134 set_level(unsigned int level);
↪→ class 135
83 * 136
84 * The first instance opens the I2C 137 private:
↪→ connection. 138 /// static pointer used to ensure a
85 * ↪→ single instance of the class
86 * @return Instance of RPiI2C 139 static RPiI2C* m_device;
87 */ 140
88 static RPiI2C& 141 /// private constructor for Singleton
89 device() 142 RPiI2C();
90 { 143
91 if (!m_device) 144 /// own logger
92 m_device = new RPiI2C(); 145 mxlog m_logger;
93 return *m_device; 146
94 } 147 /// I2C file handler
95 148 int m_fd_I2C;
96 /** 149
97 * \brief open I2C connection 150 };
98 * 151
152
153 #endif // RPII2C_H
Code C.9: Source file of the C++ class for the I 2C communication of the SiDet
device driver.
File: RPiI2C.cpp
1 /** 9 * Source: https://raspberry-projects.com/ c
2 * \file RPiI2C.cpp ↪→ pi/programming-in-c/i2c/using-the-i2c- c
3 * ↪→ interface
4 * \author Stefan Lunkenheimer 10 */
5 * Contact: stelunke@uni-mainz.de 11
6 * 12 #include "RPiI2C.h"
7 * Created on: Thu Feb 27th, 2020 13 #include "spdlog/logger.h"
8 * 14 #include <stdio.h>
148
Appendix C - EPICS Support Files
15 #include <iostream> 71 {
16 #include <unistd.h> //Needed for I2C 72 std::string s = "";
↪→ port 73 for (unsigned int i=0; i<len;
17 #include <fcntl.h> //Needed for I2C ↪→ i++)
↪→ port 74 s = fmt::format("{}{: c
18 #include <sys/ioctl.h> //Needed for I2C ↪→ #04x} ",s,
↪→ port ↪→ buffer[i]);
19 #include <linux/i2c-dev.h> //Needed for I2C 75 m_logger.trace("{:#04x} read
↪→ port ↪→ {}: {}", slaveAddr, len,
20 ↪→ s);
21 RPiI2C* RPiI2C::m_device = nullptr; 76 }
22 77 return true;
23 RPiI2C::RPiI2C(): 78 }
24 m_logger(this->IsA()), 79 else
25 m_fd_I2C(-1) 80 {
26 { 81 m_logger.error("Unable to read
27 this->open(); ↪→ from slave {:#04x}",
28 m_logger.debug("constructed"); ↪→ slaveAddr);
29 } 82 }
30 83 }
31 RPiI2C::~RPiI2C() 84 else
32 { 85 m_logger.error("Failed to acquire bus
33 this->close(); ↪→ access and/or talk to slave
34 delete m_device; ↪→ {:#04x}", slaveAddr);
35 m_logger.debug("deconstructed"); 86
36 } 87 return false;
37 88 }
38 bool RPiI2C::writeI2C(uint8_t slaveAddr, 89
↪→ uint8_t* buffer, uint8_t len) 90 bool RPiI2C::getStatus() const
39 { 91 {
40 if (ioctl(m_fd_I2C, I2C_SLAVE, slaveAddr) 92 if (m_fd_I2C >= 0)
↪→ >= 0) 93 return true;
41 { 94 return false;
42 if (write(m_fd_I2C, buffer, len) == 95 }
↪→ len) 96
43 { 97 std::string RPiI2C::IsA() const
44 if (m_logger.get_level() == 98 {
↪→ spdlog::level::trace) 99 return "RPiI2C";
45 { 100 }
46 std::string s = ""; 101
47 for (unsigned int i=0; i<len; 102 bool RPiI2C::open()
↪→ i++) 103 {
48 s = fmt::format("{}{: c 104 if (getStatus())
↪→ #04x} ",s, 105 {
↪→ buffer[i]); 106 m_logger.warn("Connection already
49 m_logger.trace("{:#04x} write ↪→ open, try reopen()");
↪→ {}: {}", slaveAddr, len, 107 return true;
↪→ s); 108 } else if ((m_fd_I2C =
50 } ↪→ ::open("/dev/i2c-1", O_RDWR)) >= 0)
51 return true; 109 {
52 } 110 m_logger.info("Connection to bus {}
53 else ↪→ opened", "/dev/i2c-1");
54 { 111 return true;
55 m_logger.error("Unable to write 112 }
↪→ to slave: {:#04x}", 113
↪→ slaveAddr); 114 m_logger.error("Failed to open the i2c
56 } ↪→ bus {}", "/dev/i2c-1");
57 } 115 return false;
58 else 116 }
59 m_logger.error("Failed to acquire bus 117
↪→ access and/or talk to slave 118 bool RPiI2C::reopen()
↪→ {:#04x}", slaveAddr); 119 {
60 120 return close() && open();
61 return false; 121 }
62 } 122
63 123 bool RPiI2C::close()
64 bool RPiI2C::readI2C(uint8_t slaveAddr, 124 {
↪→ uint8_t* buffer, uint8_t len) 125 if (m_fd_I2C >= 0)
65 { 126 {
66 if (ioctl(m_fd_I2C, I2C_SLAVE, slaveAddr) 127 ::close(m_fd_I2C);
↪→ >= 0) 128 m_fd_I2C = -1;
67 { 129 m_logger.info("Connection to bus {}
68 if (read(m_fd_I2C, buffer, len) == ↪→ closed", "/dev/i2c-1");
↪→ len) 130 }
69 { 131 return true;
70 if (m_logger.get_level() ==
↪→ spdlog::level::trace)
149
Appendix C - EPICS Support Files
132 } 138 }
133 139
134 void 140 void
135 RPiI2C::set_level(spdlog::level::level_enum 141 RPiI2C::set_level(unsigned int level)
↪→ level) 142 {
136 { 143 m_logger.set_level(level);
137 m_logger.set_level(level); 144 }
Code C.10: Header file of the C++ class for the mxlog class for the SiDet
device driver.
File: mxlog.h
1 /** 54 * @param type logger name for
2 * \class mxlog ↪→ identification, should be unique
3 * 55 */
4 * \brief mxlogger 56 mxlog(spdlog::level::level_enum level,
5 * ↪→ std::string type = "undefined");
6 * This class allows to specify seperat 57
↪→ loggers per object. The advantage is, 58 /**
7 * that the log_level for each logger can set 59 * \brief default constructor
↪→ seperately. 60 *
8 * 61 * @param level initial logger level
9 * \todo Add file_logger 62 * @param type logger name for
10 * \todo Think about this class, maybe it's ↪→ identification, should be unique
↪→ better derivate from this class 63 */
11 * \todo How to dis/enable spdlog? 64 mxlog(unsigned int level, std::string
12 * ↪→ type = "undefined");
13 * \author Stefan Lunkenheimer 65
14 * Contact: stelunke@uni-mainz.de 66 /// default destructor
15 * 67 ~mxlog();
16 * Created on: Thu Mar 12th, 2020 68
17 * 69 /**
18 */ 70 * \brief log string with fmt formating
19 ↪→ @ref format with severity @ref level
20 #ifndef MXLOG_H 71 *
21 #define MXLOG_H 72 * @param level log level
22 73 * @param format fmt format string
23 #include <memory> 74 * @args the arguments used to format the
24 #include <iostream> ↪→ string @ref format
25 75 */
26 #include "spdlog/spdlog.h" 76 template<typename ... Args>
27 #include "spdlog/sinks/stdout_color_sinks.h" 77 void log(int level, const std::string&
28 #include "spdlog/sinks/basic_file_sink.h" ↪→ format, const Args ... args )
29 78 {
30 79 m_logger->log(level, format,
31 class mxlog final ↪→ args...);
32 { 80 }
33 public: 81
34 /** 82 /**
35 * \brief default constructor 83 * \brief log string with fmt formating
36 * ↪→ @ref format with severity @ref level
37 * @param type logger name for 84 *
↪→ identification, should be unique 85 * @param level log level
38 * @param level initial logger level 86 * @param format fmt format string
39 */ 87 * @args the arguments used to format the
40 mxlog(std::string type = "undefined", ↪→ string @ref format
↪→ spdlog::level::level_enum 88 */
↪→ level=spdlog::level::info); 89 template<typename ... Args>
41 90 void log(spdlog::level::level_enum
42 /** ↪→ level, const std::string& format,
43 * \brief default constructor ↪→ const Args ... args )
44 * 91 {
45 * @param type logger name for 92 m_logger->log(level, format,
↪→ identification, should be unique ↪→ args...);
46 * @param level initial logger level 93 }
47 */ 94
48 mxlog(std::string type, unsigned int 95
↪→ level); 96 /**
49 97 * \brief log string with fmt formating
50 /** ↪→ @ref format with severity @ref level
51 * \brief default constructor 98 *
52 * 99 * @param level log level
53 * @param level initial logger level 100 * @param format fmt format string
150
Appendix C - EPICS Support Files
101 * @args the arguments used to format the 153 {
↪→ string @ref format 154 m_logger->error(format, args...);
102 */ 155 }
103 template<typename ... Args> 156
104 void trace(const std::string& format, 157 /**
↪→ const Args ... args ) 158 * \brief log string with fmt formating
105 { ↪→ @ref format with severity critical
106 m_logger->trace(format, args...); 159 *
107 } 160 * @param format fmt format string
108 161 * @args the arguments used to format the
109 /** ↪→ string @ref format
110 * \brief log string with fmt formating 162 */
↪→ @ref format with severity debug 163 template<typename ... Args>
111 * 164 void critical(const std::string& format,
112 * @param format fmt format string ↪→ const Args ... args )
113 * @args the arguments used to format the 165 {
↪→ string @ref format 166 m_logger->critical(format, args...);
114 */ 167 }
115 template<typename ... Args> 168
116 void debug(const std::string& format, 169 /**
↪→ const Args ... args ) 170 * \brief get the current log level
117 { 171 *
118 m_logger->debug(format, args...); 172 * @return current log level
119 } 173 */
120 174 spdlog::level::level_enum get_level()
121 /** ↪→ const;
122 * \brief log string with fmt formating 175
↪→ @ref format with severity info 176 /**
123 * 177 * \brief set log level
124 * @param format fmt format string 178 *
125 * @args the arguments used to format the 179 * @param log_level
↪→ string @ref format 180 */
126 */ 181 void set_level(spdlog::level::level_enum
127 template<typename ... Args> ↪→ log_level);
128 void info(const std::string& format, 182
↪→ const Args ... args ) 183 /**
129 { 184 * \brief set log level
130 m_logger->info(format, args...); 185 *
131 } 186 * @param log_level
132 187 */
133 /** 188 void set_level(unsigned int log_level);
134 * \brief log string with fmt formating 189
↪→ @ref format with severity warning 190 // void
135 * 191 // enableFile(std::string filename);
136 * @param format fmt format string 192
137 * @args the arguments used to format the 193 //void trace
↪→ string @ref format 194 private:
138 */ 195 /// current log level
139 template<typename ... Args> 196 spdlog::level::level_enum m_level;
140 void warn(const std::string& format, 197
↪→ const Args ... args ) 198 /// console sink - equal for all logger
141 { 199 static std::shared_ptr<spdlog::sinks:: c
142 m_logger->warn(format, args...); ↪→ stdout_color_sink_mt>
143 } ↪→ m_console_sink;
144 200
145 /** 201 //static std::shared_ptr<spdlog::sinks:: c
146 * \brief log string with fmt formating ↪→ basic_file_sink_mt>
↪→ @ref format with severity error ↪→ m_file_sink;
147 * 202
148 * @param format fmt format string 203 /// specific logger for each object
149 * @args the arguments used to format the 204 std::shared_ptr<spdlog::logger> m_logger;
↪→ string @ref format 205 };
150 */ 206
151 template<typename ... Args> 207 #endif //MXLOG_H
152 void error(const std::string& format,
↪→ const Args ... args )
Code C.11: Source file of the C++ class for the mxlog class for the SiDet device
driver.
File: mxlog.cpp
1 /** 4 * \author Stefan Lunkenheimer
2 * \file mxlog.cpp 5 * Contact: stelunke@uni-mainz.de
3 * 6 *
151
Appendix C - EPICS Support Files
7 * Created on: Thu Mar 12th, 2020 41
8 * 42 mxlog::mxlog(spdlog::level::level_enum
9 **/ ↪→ level, std::string type):
10 43 mxlog(type, level)
11 #include <stdio.h> 44 {
12 #include <iostream> 45 }
13 #include <unistd.h> 46
14 #include <memory> 47 mxlog::mxlog(unsigned int level, std::string
15 #include <string> ↪→ type):
16 48 mxlog(type, static_cast<spdlog::level:: c
17 #include "mxlog.h" ↪→ level_enum>(level))
18 49 {
19 50
20 std::shared_ptr<spdlog::sinks:: c 51 }
↪→ stdout_color_sink_mt> 52
↪→ mxlog::m_console_sink = nullptr; 53 mxlog::~mxlog()
21 //std::shared_ptr<spdlog::sinks:: c 54 {
↪→ basic_file_sink_mt> 55 m_logger->info("Logger \"{}\" stopped",
↪→ mxlog::m_file_sink=nullptr; ↪→ m_logger->name());
22 56 m_logger->flush();
23 mxlog::mxlog(std::string type, 57 m_logger.reset();
↪→ spdlog::level::level_enum level): 58 }
24 m_level(level), 59
25 m_logger(nullptr) 60 spdlog::level::level_enum mxlog::get_level()
26 { ↪→ const {
27 if (!m_console_sink) 61 return m_level;
28 m_console_sink = 62 }
↪→ std::make_shared<spdlog::sinks:: c 63
↪→ stdout_color_sink_mt>(); 64 void mxlog::set_level(spdlog::level:: c
29 ↪→ level_enum log_level)
30 m_logger = std::make_shared<spdlog:: c ↪→ {
↪→ logger>(type, 65 m_level = log_level;
↪→ m_console_sink); 66 m_logger->info("set_level to {}",
31 m_logger->set_level(m_level); ↪→ spdlog::level:: c
32 log(spdlog::level::info, "Logger \"{}\" ↪→ to_string_view(log_level));
↪→ created", type); 67 m_logger->set_level(m_level);
33 } 68 }
34 69
35 mxlog::mxlog(std::string type, unsigned int 70 void mxlog::set_level(unsigned int log_level)
↪→ level): ↪→ {
36 mxlog(type, static_cast<spdlog::level:: c 71 if (log_level < spdlog::level::n_levels)
↪→ level_enum>(level)) 72 set_level(static_cast<spdlog:: c
37 { ↪→ level::level_enum>(log_level));
38 } 73 else
39 74 set_level(spdlog::level::off);
40 75 }
Code C.12: The read_aiSIDET and write_aiSIDET function defined to con-
nect the SiDet library with EPICS using the device support interface.
File: SILICONDETECTOR.cpp
1 #include "aiRecord.h" 23 err_msg (std::string recName, std::string e,
2 #include "aoRecord.h" ↪→ std::string pattern = "\u274C%s
3 #include "epicsExport.h" ↪→ \u25AC\u25AC\u25B6 %s!!!")
4 #include "iocsh.h" 24 {
5 #include <vector> 25 char msg[MAX_MSG_LENGTH];
6 #include <algorithm> // std::find_if 26 sprintf(msg, pattern.c_str(),
7 #include <string> ↪→ recName.c_str(), e.c_str());
8 #include <unistd.h> 27 std::cerr << "\033[1;31m" << msg <<
9 #include <iostream> ↪→ "\033[0m" << std::endl;
10 28 }
11 #include "SiDet.h" 29
12 #include "RPiI2C.h" 30 extern "C" long
13 31 init_recordSIDET (struct aiRecord *pai)
14 #include <mutex> // std::mutex 32 {
15 33 std::lock_guard<std::mutex>
16 #define MAX_MSG_LENGTH 100 ↪→ lock(m_AccessMutex);
17 34 if (!mySidet)
18 static std::mutex m_AccessMutex; 35 {
19 36 mySidet = new SiDet();
20 static SiDet * mySidet = nullptr; 37 }
21 38 return ( 0 );
22 static void 39 }
40
152
Appendix C - EPICS Support Files
41 /** 102 if
42 * ↪→ (pString[0].compare("getResistance")
43 * @param pai ↪→ == 0)
44 * @return error code: 2 := success in 103 {
↪→ engineering units; 3:= Device not found 104 if (pString.size() < 2)
45 */ 105 {
46 extern "C" long 106 err_msg(pai->name, "Not
47 read_aiSIDET (struct aiRecord *pai) ↪→ Enough Parameters");
48 { 107 return ( 3 );
49 if (pai->inp.text) 108
50 { 109 }
51 // Split input String 110 std::uint8_t channel =
52 std::string input { pai->inp.text }; ↪→ atoi(pString[1].c_str());
53 std::string delimiter = " "; 111 if (channel >= 2)
54 std::vector < std::string > pString; 112 {
55 113 err_msg(pai->name, "Channel
56 size_t pos = 0; ↪→ Out Of Range");
57 std::string token; 114 return ( 3 );
58 while (( pos = input.find(delimiter) 115 }
↪→ ) != std::string::npos) 116 double value = 0;
59 { 117 std::lock_guard<std::mutex>
60 token = input.substr(0, pos); ↪→ lock(m_AccessMutex);
61 pString.push_back(token); 118 if (!(mySidet- c
62 input.erase(0, pos + ↪→ >getResistance(&value,
↪→ delimiter.length()); ↪→ channel)))
63 } 119 {
64 pString.push_back(input); 120 err_msg(pai->name, "Can't
65 ↪→ Read Resistance");
66 // Check number of input parameters 121 return ( 3 );
67 if (pString.size() == 0) 122 }
68 { 123 pai->val = value;
69 err_msg(pai->name, "Not Enough 124 pai->udf = FALSE;
↪→ Parameters"); 125 return ( 2 );
70 return ( 3 ); 126 }
71 127 }
72 } 128 else
73 if 129 {
↪→ (pString[0].compare("getThreshold")130 err_msg(pai->name, "No input");
↪→ == 0) 131 return ( 3 );
74 { 132 }
75 double value = 0; 133 err_msg(pai->name, "Unknowm Command");
76 std::lock_guard<std::mutex> 134 return ( 3 );
↪→ lock(m_AccessMutex); 135 }
77 if (!(mySidet- c 136
↪→ >getThreshold(&value))) 137 extern "C" long
78 { 138 write_aoSIDET (struct aoRecord *pai)
79 err_msg(pai->name, "Can't 139 {
↪→ Read Threshold"); 140 if (pai->out.text)
80 return ( 3 ); 141 {
81 } 142 // Split input String
82 pai->val = value; 143 std::string input { pai->out.text };
83 pai->udf = FALSE; 144 std::string delimiter = " ";
84 return ( 2 ); 145 std::vector < std::string > pString;
85 } 146
86 147 size_t pos = 0;
87 if 148 std::string token;
↪→ (pString[0].compare("getTemperature1"4)9 while (( pos = input.find(delimiter)
↪→ == 0) ↪→ ) != std::string::npos)
88 { 150 {
89 double value = 0; 151 token = input.substr(0, pos);
90 std::lock_guard<std::mutex> 152 pString.push_back(token);
↪→ lock(m_AccessMutex); 153 input.erase(0, pos +
91 if (!(mySidet- c ↪→ delimiter.length());
↪→ >getPCBTemperature(&value))) 154 }
92 { 155 pString.push_back(input);
93 err_msg(pai->name, "Can't 156
↪→ Read getTemperature"); 157 // Check number of input parameters
94 return ( 3 ); 158 if (pString.size() == 0)
95 } 159 {
96 pai->val = value; 160 err_msg(pai->name, "Not Enough
97 pai->udf = FALSE; ↪→ Parameters");
98 return ( 2 ); 161 return ( 3 );
99 } 162
100 163 }
101 // getResistance Readout 164 if
↪→ (pString[0].compare("setThreshold")
↪→ == 0)
153
Appendix C - EPICS Support Files
165 { 173 pai->udf = FALSE;
166 std::lock_guard<std::mutex> 174 return ( 0 );
↪→ lock(m_AccessMutex); 175 }
167 double value = (double) pai->val; 176 }
168 if (!(mySidet- c 177 else
↪→ >setThreshold(value))) 178 {
169 { 179 err_msg(pai->name, "No input");
170 err_msg(pai->name, "Can't Set 180 return ( 3 );
↪→ Threshold"); 181 }
171 return ( 3 ); 182 err_msg(pai->name, "Unknowm Command");
172 } 183 return ( 3 );
184 }
Code C.13: The definition of the IOC shell command MXSiDet_SetLogLevel.
File: SILICONDETECTOR.cpp
1 /* 45 << "Usage: MXSiDet_SetLogLevel c
2 struct iocshFuncDef { ↪→ (s,d='all')" <<
3 const char *name; ↪→ std::endl
4 int nargs; 46 << "Set the minimal Message
5 const iocshArg * const *arg; ↪→ log_level s for device d"
6 }; ↪→ << std::endl
7 47 << "string s: \"trace\",
8 struct iocshArg { ↪→ \"debug\", \"info\",
9 const char *name; ↪→ \"warning\", \"error\",
10 iocshArgType type; ↪→ \"critical\", \"off\"" <<
11 }iocshArg; ↪→ std::endl
12 48 << "string d: \"all\"
13 void showCallFunc(const iocshArgBuf *args); ↪→ (including: \"SiDet\",
14 ↪→ \"MUX\", \"DAC\", \"ADC\",
15 iocshArgInt int args[i].ival ↪→ \"PCB\") or \"I2C\""
16 iocshArgDouble double args[i].dval 49 << "\033[0m" << std::endl;
17 iocshArgString char * args[i].sval ↪→ //back to standard
18 iocshArgPersistentString char * args[i].sval 50 }
19 iocshArgPdbbase void * args[i].vval 51
20 iocshArgArgv int args[i].aval.ac 52 void SetMXSiDetLogLevelCallFunc(const
21 char ** args[i].aval.av ↪→ iocshArgBuf *args)
22 */ 53 {
23 54
24 /* drvXxx code, FuncDef and CallFunc 55
↪→ definitions ... */ 56 if (args[0].sval==NULL ||
25 ↪→ args[0].sval[0]=='?')
26 static const iocshArg SetMXSiDetLogLevelArg0 57 {
↪→ = { "log_level", iocshArgString }; 58 printDoc();
27 static const iocshArg SetMXSiDetLogLevelArg1 59 return;
↪→ = { "device", iocshArgString }; 60 }
28 61
29 static const iocshArg *const 62 std::string arg0{args[0].sval};
↪→ SetMXSiDetLogLevelArgs[2] = 63 spdlog::level::level_enum log_level =
↪→ {&SetMXSiDetLogLevelArg0, c ↪→ spdlog::level::from_str(arg0);
↪→ &SetMXSiDetLogLevelArg1}; 64 if (log_level == spdlog::level::off
30 static const iocshFuncDef 65 && arg0.compare("off") != 0)
↪→ SetMXSiDetLogLevelDef = 66 {
↪→ {"MXSiDet_SetLogLevel",2, c 67 char msg[50];
↪→ SetMXSiDetLogLevelArgs}; 68 sprintf(msg,"Severity Level %s
31 ↪→ unknown",arg0.c_str());
32 static void SetMXSiDetLogLevelCallFunc(const 69 err_msg("MXSiDet_SetLogLevel", msg);
↪→ iocshArgBuf *args); 70 printDoc();
33 71 return;
34 /* drvXxx registration Function and export 72 }
*/ 73
35 static void SetMXSiDetLogLevelRegistrar(void) 74 std::lock_guard<std::mutex>
36 { ↪→ lock(m_AccessMutex);
37 iocshRegister(&SetMXSiDetLogLevelDef, 75 if (!mySidet)
↪→ SetMXSiDetLogLevelCallFunc); 76 {
38 } 77 mySidet = new SiDet();
39 epicsExportRegistrar c 78 }
↪→ (SetMXSiDetLogLevelRegistrar); 79 if (args[1].sval==NULL)
40 /* in dbd add registrar(drvXxxRegistrar) */ 80 {
41 81 mySidet->set_level(log_level,true);
42 static void printDoc() 82 return;
43 { 83 }
44 std::cout << "\033[1;36m" //cyan 84
85 std::string arg1{args[1].sval};
86 if(arg1.compare("all") == 0)
154
Appendix C - EPICS Support Files
87 { 105 }
88 mySidet->set_level(log_level,true); 106 else if (arg1.compare("ADC") == 0)
89 return; 107 {
90 } 108 mySidet->set_level(log_level,SiDet:: c
91 else if (arg1.compare("SiDet") == 0) ↪→ ADC);
92 { 109 return;
93 mySidet->set_level(log_level,false); 110 }
94 return; 111 else if (arg1.compare("PCB") == 0)
95 } 112 {
96 else if (arg1.compare("MUX") == 0) 113 mySidet->set_level(log_level,SiDet:: c
97 { ↪→ PCB);
98 mySidet->set_level(log_level,SiDet:: c 114 return;
↪→ MUX); 115 }
99 return; 116 else if (arg1.compare("I2C") == 0)
100 } 117 {
101 else if (arg1.compare("DAC") == 0) 118 RPiI2C::device().set_level c
102 { ↪→ (log_level);
103 mySidet->set_level(log_level,SiDet:: c 119 return;
↪→ DAC); 120 }
104 return; 121 }
155
156
APPENDIX D
Supplemental Material –
Simulation
In this chapter the definition of the TTree structure and the configuration files for
the simulation are given.
Code D.1: The TTree structure for the particle and event, stored in a root file
at the end of the simulation.
File: SFactorFullAnalysisTTree.hh
1 struct Particle{ 9
2 Double_t 10 struct Event {
3 vertex_x,vertex_y,vertex_z,mass, 11 Double_t
4 E,pp,p_phi,p_theta,p_ctheta; 12 weight,
5 Int_t PID; 13 W, Phi, q2,
6 }m_IS_electron, m_IS_oxygen, 14 CMS_theta, CMS_phi;
↪→ m_FS_electron, m_FS_particle1, 15 Int_t nr = 0;
↪→ m_FS_particle2, m_ImS_oxygen, 16 }m_event;
↪→ m_ImS_vPhoton; 17
7 18 const char *TTreestructureEvent =
8 const char *TTreestructureParticle = ↪→ "weight/D:W:Phi:1_q2:CMT_theta: c
↪→ "vertex_x/D:vertex_y:vertex_z:mass: c ↪→ CMS_phi:nr/I";
↪→ E:pp:p_phi:p_theta:p_ctheta:PID/I";
Code D.2: Main configuration file of the simulation.
File: SFactor_CrossSectionIntegrator_ZeroDegTagger.xml
1 <magix> 13 <path>@MAGIX_DIR@/lib/MXWare/ c
2 ↪→ plugins</path>
3 <!-- Enabling the logging system --> 14 <printlist>true</printlist>
4 <logging> 15 <stopaterr>false</stopaterr>
5 <logfile>./log/MXLog.txt</logfile> 16 </pluginmanager>
6 <console>true</console> 17
7 <severity>info</severity> 18 <plugin pluginname="MXUtilityTools">
8 </logging> 19 <pdatatable>@CMAKE_INSTALL_PREFIX@/ c
9 ↪→ share/MXWare/myPartData.mcd</ c
10 <!-- Configuring the plugin manager ↪→ pdatatable>
↪→ --> 20 <LogPDataTable>true</LogPDataTable>
11 <pluginmanager> 21 </plugin>
12 <path>@CMAKE_INSTALL_PREFIX@/lib/ c 22
↪→ MXWare/plugins</path> 23 <plugin pluginname= c
↪→ "CrossSectionIntegrator">
157
Appendix D - Supplemental Material – Simulation
24 <threads>8</threads> 66 <E1data>@CMAKE_INSTALL_PREFIX@/ c
25 <engine type="NR_HQGen"/> ↪→ share/MXWare/SE1data.csv</ c
26 <autoseed>true</autoseed> ↪→ E1data>
27 <samples>20000000</samples> 67 <E2data>@CMAKE_INSTALL_PREFIX@/ c
28 <accuracy>0.001</accuracy> ↪→ share/MXWare/SE2data.csv</ c
29 <beam type= c ↪→ E2data>
↪→ "IdealBeamParticleGenerator"> 68 </model>
30 <momentum>105</momentum> 69
31 <center>(0, 0)</center> 70 <full_analysis type="SFactor_ c
32 <pid>11</pid> ↪→ FullAnalysisTTree">
33 <current>1.0</current> 71 <rootfile>
34 <helicity>0.2</helicity> 72 <path>@CMAKE_INSTALL_PREFIX@/ c
35 </beam> ↪→ out/SFactor_TTree_ c
36 <beam_analysis ↪→ ZeroDeg.root</path>
↪→ type="InitialBeamRootSummary"> ↪→
↪→ 73 <openmode>RECREATE</openmode>
37 <rootfile> ↪→
38 <path>@CMAKE_INSTALL_PREFIX@/ c 74 </rootfile>
↪→ out/SFactor_ZeroDeg.root</ c 75 <rootfolder>SFactor</rootfolder>
↪→ path> 76 </full_analysis>
↪→ 77
39 <openmode>RECREATE</openmode> 78 <finalstate_analysis
↪→ ↪→ type="SFactor_FSRootAnalysis">
40 </rootfile> ↪→
41 <rootfolder>SFactor/InitialBeam/ c 79 <rootfile>
↪→ </rootfolder> 80 <path>@CMAKE_INSTALL_PREFIX@/ c
↪→ ↪→ out/SFactor_ZeroDeg.root</ c
42 </beam_analysis> ↪→ path>
43 <target type= c ↪→
↪→ "TransverseUniformLayer_TIG"> 81 <openmode>UPDATE</openmode>
44 <density>0.21219</density> <!-- ↪→
↪→ 2.1219*10^18 particles / 82 </rootfile>
↪→ cm^2--> 83 <rootfolder>SFactor_FSG</ c
45 <x_extension>[-10:10]</x_ c ↪→ rootfolder>
↪→ extension> 84 <seconds>2419200</seconds> <!-- 4
46 <y_extension>[-10:10]</y_ c ↪→ weeks -->
↪→ extension> 85 <histo_CMSEnergy>
47 <z_extension>[-1:1]</z_extension> 86 <name>CMS_Energy</name>
48 <pid>1000080160</pid> 87 <title>Center of Mass Energy of
49 </target> ↪→ the outgoing
50 <initialstate_analysis ↪→ particles</title>
↪→ type="InitialStateRootSummary"> 88 <nxbins>200</nxbins>
↪→ 89 <xmin>0.2</xmin>
51 <rootfile> 90 <xmax>14</xmax>
52 <path>@CMAKE_INSTALL_PREFIX@/ c 91 </histo_CMSEnergy>
↪→ out/SFactor_ZeroDeg.root</ c 92 <histo_q2>
↪→ path> 93 <name>q2</name>
↪→ 94 <title>Momentum transfer of the
53 <openmode>UPDATE</openmode> ↪→ scattered electron to the
↪→ ↪→ target</title>
54 </rootfile> 95 <nxbins>100</nxbins>
55 <rootfolder>SFactor/InitialState/ c 96 <xmin>-1e-6</xmin>
↪→ </rootfolder> 97 <xmax>0</xmax>
56 </initialstate_analysis> 98 </histo_q2>
57 <final_state type="Inelastic_ c 99 <histo_q2Inverse>
↪→ TwoBodyGenerator_FSG"> 100 <name>Inverse q2</name>
58 <PID1>1000020040</PID1> 101 <title>Inverse Momentum transfer
59 <PID2>1000060120</PID2> ↪→ of the scattered electron to
60 <name>TwoBodyGenerator</name> ↪→ the target</title>
61 <cutoff>0</cutoff> 102 <nxbins>200</nxbins>
62 <dWmax>25</dWmax> 103 <xmin>-15e3</xmin>
63 </final_state> 104 <xmax>-1e3</xmax>
64 <model type= c 105 </histo_q2Inverse>
↪→ "SFactorCrossSectionCalculator"> 106 <hist_thetavsTkinScat>
65 <structure>extrap</structure> 107 <nxbins>320</nxbins>
108 <xmin>-0.0001</xmin>
158
Appendix D - Supplemental Material – Simulation
109 <xmax>0.005</xmax> 121 </histo_thetaScat>
110 <nybins>320</nybins> 122
111 <ymin>55</ymin> 123 </finalstate_analysis>
112 <ymax>100</ymax> 124
113 </hist_thetavsTkinScat> 125 <_include>@CMAKE_INSTALL_PREFIX@/ c
114 ↪→ share/MXWare/examples/TwoBody_ c
115 <histo_thetaScat> ↪→ DetectorSetup_ZeroDegTagger.xml c
116 <name>theta_scattered</name> ↪→ </_include>
117 <title>The azimuthal angle of 126
↪→ the direction of the 127 </plugin>
↪→ scattered particle</title> 128
118 <nxbins>320</nxbins> 129
119 <xmin>-0.0001</xmin> 130 </magix>
120 <xmax>0.015</xmax>
Code D.3: Detector setup used in the simulation for Phase 1
File: TwoBody_DetectorSetup_Spectrometer.xml
1 <acceptance 34 </acceptance>
↪→ type="CoincidenceAcceptance"> 35
2 <acceptance type= c 36 <acceptance type= c
↪→ "RotatingSpectrometerAcceptance"> ↪→ "SiliconStripAcceptance">
3 <acceptV>true</acceptV> 37 <name>SiliconStripDetector3</name>
4 <acceptancefile>@MAGIX_DIR@/@DATA_ c 38 <distance>100</distance>
↪→ INSTALL_DIR@/MXSpectrometer_ c 39 <rotation_angle_deg>150 c
↪→ 2013.dat</acceptancefile> ↪→ </rotation_angle_deg>
5 <central_momentum>-75</central_ c 40 <acceptV>true</acceptV>
↪→ momentum> 41 <VBlockSize>10</VBlockSize>
6 <distance>700</distance> 42 <width>50</width>
7 <name/> 43 <length>50</length>
8 <offplane_angle>0</offplane_angle> 44 <PIDRestrict>1000020040 c
9 <offsetz>0</offsetz> ↪→ </PIDRestrict>
10 <rotation_angle_deg>-10</rotation_ c 45 </acceptance>
↪→ angle_deg> 46
11 </acceptance> 47 <acceptance type= c
12 ↪→ "SiliconStripAcceptance">
13 <acceptance type="AcceptanceSet"> 48 <name>SiliconStripDetector5</name>
14 <acceptance type= c 49 <distance>100</distance>
↪→ "SiliconStripAcceptance"> 50 <rotation_angle_deg>-90 c
15 <name>SiliconStripDetector1</name> ↪→ </rotation_angle_deg>
16 <distance>100</distance> 51 <acceptV>true</acceptV>
17 <rotation_angle_deg>30</rotation_ c 52 <VBlockSize>10</VBlockSize>
↪→ angle_deg> 53 <width>50</width>
18 <acceptV>true</acceptV> 54 <length>50</length>
19 <VBlockSize>10</VBlockSize> 55 <PIDRestrict>1000020040 c
20 <width>50</width> ↪→ </PIDRestrict>
21 <length>50</length> 56 </acceptance>
22 <PIDRestrict>1000020040 c 57
↪→ </PIDRestrict> 58 <acceptance type= c
23 </acceptance> ↪→ "SiliconStripAcceptance">
24 59 <name>SiliconStripDetector6</name>
25 <acceptance 60 <distance>100</distance>
↪→ type="SiliconStripAcceptance"> 61 <rotation_angle_deg>-150 c
26 <name>SiliconStripDetector2</name> ↪→ </rotation_angle_deg>
27 <distance>100</distance> 62 <acceptV>true</acceptV>
28 <rotation_angle_deg>90</rotation_ c 63 <VBlockSize>10</VBlockSize>
↪→ angle_deg> 64 <width>50</width>
29 <acceptV>true</acceptV> 65 <length>50</length>
30 <VBlockSize>10</VBlockSize> 66 <PIDRestrict>1000020040 c
31 <width>50</width> ↪→ </PIDRestrict>
32 <length>50</length> 67 </acceptance>
33 <PIDRestrict>1000020040 c 68 </acceptance>
↪→ </PIDRestrict> 69 </acceptance>
159
Appendix D - Supplemental Material – Simulation
Code D.4: Detector setup used in the simulation for Phase 2
File: TwoBody_DetectorSetup_ZeroDegTagger.xml
1 <acceptance 39 <PIDRestrict>1000020040</ c
↪→ type="CoincidenceAcceptance"> ↪→ PIDRestrict>
2 40 </acceptance>
3 <acceptance type= c 41
↪→ "ZeroDegreeSpectrometerAcceptance"> 42 <acceptance
4 <momentumRange>[7:105]</ c ↪→ type="SiliconStripAcceptance">
↪→ momentumRange> 43 <name>SiliconStripDetector4</name>
5 <thetaRange_deg>[0:0.85]</ c 44 <distance>100</distance>
↪→ thetaRange_deg> 45 <rotation_angle_deg>-30</ c
6 </acceptance> ↪→ rotation_angle_deg>
7 46 <acceptV>true</acceptV>
8 <acceptance type="AcceptanceSet"> 47 <VBlockSize>10</VBlockSize>
9 <acceptance type= c 48 <width>50</width>
↪→ "SiliconStripAcceptance"> 49 <length>50</length>
10 <name>SiliconStripDetector1</name> 50 <PIDRestrict>1000020040</ c
11 <distance>100</distance> ↪→ PIDRestrict>
12 <rotation_angle_deg>30</ c 51 <PIDRestrict>1000060120</ c
↪→ rotation_angle_deg> ↪→ PIDRestrict>
13 <acceptV>true</acceptV> 52 </acceptance>
14 <VBlockSize>10</VBlockSize> 53
15 <width>50</width> 54 <acceptance type= c
16 <length>50</length> ↪→ "SiliconStripAcceptance">
17 <PIDRestrict>1000020040</ c 55 <name>SiliconStripDetector5</name>
↪→ PIDRestrict> 56 <distance>100</distance>
18 </acceptance> 57 <rotation_angle_deg>-90</ c
19 ↪→ rotation_angle_deg>
20 <acceptance 58 <acceptV>true</acceptV>
↪→ type="SiliconStripAcceptance"> 59 <VBlockSize>10</VBlockSize>
21 <name>SiliconStripDetector2</name> 60 <width>50</width>
22 <distance>100</distance> 61 <length>50</length>
23 <rotation_angle_deg>90</ c 62 <PIDRestrict>1000020040</ c
↪→ rotation_angle_deg> ↪→ PIDRestrict>
24 <acceptV>true</acceptV> 63 </acceptance>
25 <VBlockSize>10</VBlockSize> 64
26 <width>50</width> 65 <acceptance type= c
27 <length>50</length> ↪→ "SiliconStripAcceptance">
28 <PIDRestrict>1000020040</ c 66 <name>SiliconStripDetector6</name>
↪→ PIDRestrict> 67 <distance>100</distance>
29 </acceptance> 68 <rotation_angle_deg>-150</ c
30 ↪→ rotation_angle_deg>
31 <acceptance type= c 69 <acceptV>true</acceptV>
↪→ "SiliconStripAcceptance"> 70 <VBlockSize>10</VBlockSize>
32 <name>SiliconStripDetector3</name> 71 <width>50</width>
33 <distance>100</distance> 72 <length>50</length>
34 <rotation_angle_deg>150</ c 73 <PIDRestrict>1000020040</ c
↪→ rotation_angle_deg> ↪→ PIDRestrict>
35 <acceptV>true</acceptV> 74 </acceptance>
36 <VBlockSize>10</VBlockSize> 75 </acceptance>
37 <width>50</width> 76 </acceptance>
38 <length>50</length>
160
List of Figures
2.1 Schematic view of a scattering process . . . . . . . . . . . . . . . 7
2.2 Maxwell-Boltzmann distribution for specific elements at the tem-
peratures for specific burning stages in stars . . . . . . . . . . . . 10
2.3 Sketch of the Gamow peak, Maxwell-Boltzmann distribution and
Coulomb penetration factor . . . . . . . . . . . . . . . . . . . . . 13
2.4 Schematic view of the Proton-Proton I-III chains . . . . . . . . . 14
2.5 Schematic view of the CNO cylce . . . . . . . . . . . . . . . . . . 16
2.6 Reaction scheme of the two most important reactions of the helium
burning stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.7 Level scheme of the nuclei involed in the helium burning stage . . 19
2.8 Level diagram of the 16O nucleus . . . . . . . . . . . . . . . . . . 20
2.9 The total S-factor compared to the ground state and cascade tran-
sitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.10 The E1 and E2 ground state contribution compared to the contri-
bution of cascade transitions . . . . . . . . . . . . . . . . . . . . . 23
2.11 Coulomb phase shift cos(Φ12) . . . . . . . . . . . . . . . . . . . . 23
2.12 Angular contribution of E1 and E2 for the ground state transition 26
2.13 Cross section of the time-reversed reaction 16O(γ, α)12C and direct
reaction 12C(α, γ)16O in comparison . . . . . . . . . . . . . . . . . 28
2.14 E1 and E2 contribution data . . . . . . . . . . . . . . . . . . . . . 33
2.15 Production factor of some key isotopes as function of the S-factor 35
3.1 First order Feynman diagram for the electro-disintegration reaction
16O(e, e′α)12C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.1 Overview of the structure of MESA . . . . . . . . . . . . . . . . . 53
4.2 Overview of the MESA hall and the experiments . . . . . . . . . . 55
4.3 Overview of the P2 experiment . . . . . . . . . . . . . . . . . . . 55
4.4 Overview of the DarkMESA experiment . . . . . . . . . . . . . . 56
4.5 Overview of the MAGIX experiment . . . . . . . . . . . . . . . . 57
4.6 Laval Nozzle and Catcher . . . . . . . . . . . . . . . . . . . . . . 58
4.7 Overview of the gas jet target . . . . . . . . . . . . . . . . . . . . 59
4.8 Overview of the MAGIX spectrometer . . . . . . . . . . . . . . . 60
4.9 Schematic principle of a TPC . . . . . . . . . . . . . . . . . . . . 61
4.10 The current 3D concept of the trigger veto system . . . . . . . . . 62
4.11 Schematic view of the cross section of the diamond-shaped seg-
mented trigger layer . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.12 Schematic view of the MAGIX trigger veto system . . . . . . . . . 63
4.13 Prototype of the silicon strip detetor . . . . . . . . . . . . . . . . 64
161
List of Figures
4.14 Schematic representation of the working principle of the silicon strip
detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.15 Schematic representation of the zero degree tagger for MAGIX . . 66
4.16 Schematic view of the experimental setup for Phase 0/1 . . . . . . 69
4.17 Schematic view of the experimental setup for Phase 2 . . . . . . . 70
5.1 Overview of the major sotware components of an IOC . . . . . . . 73
5.2 Overview of the task distribution for the IOC and the CACs . . . 77
5.3 Overview of the channel access process . . . . . . . . . . . . . . . 78
5.4 Example of a CSS OPI . . . . . . . . . . . . . . . . . . . . . . . . 82
5.5 General structure of the SiDet test stand from the point of view of
the control software . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.6 Overview of the SiDet readout electronics . . . . . . . . . . . . . . 98
5.7 Class diagram MXSiDet . . . . . . . . . . . . . . . . . . . . . . . 99
5.8 Example output of the mxlog class for the SiDet device driver . . 100
5.10 CSS OPI for the silicon strip detector test stand . . . . . . . . . . 102
6.1 Illustration of the distribution of the random numbers using a MC
integration to calculate π . . . . . . . . . . . . . . . . . . . . . . . 106
6.2 Example of a less suitable function for the use of MC integration . 108
6.3 Separation of the Feynman diagram of the electro-disintegration
reaction of 16O into two parts . . . . . . . . . . . . . . . . . . . . 113
6.4 Histogram representing the weighted cms energy Ecms for Phase 2 119
6.5 Hitmap for the silicon strip detector array . . . . . . . . . . . . . 120
6.6 Number of expected events distributed over the cms angular of the
α-particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.7 Histogram representing the isotropically distributed cms angle of
the α-particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.8 Histogram of the weighted cms angle of the α-particle . . . . . . . 122
6.9 E1 and E2 simulation results . . . . . . . . . . . . . . . . . . . . . 124
162
List of Tables
2.1 S-factor results at Ecms = 300 keV of previous works . . . . . . . . 32
5.1 Overview of the ICs placed on the SiDet frontend-board . . . . . 98
6.1 Comparison of MC integration for computing π with a pseudo-
random and a quasi-random generator . . . . . . . . . . . . . . . 107
6.2 Results of a MC integration for a less suitable function . . . . . . 109
6.3 Summary of the relevant simulation and analysis parameters. . . 123
B.1 List of adopted levels for 8Be . . . . . . . . . . . . . . . . . . . . . 135
B.2 List of adopted levels for 12C . . . . . . . . . . . . . . . . . . . . . 135
B.3 List of adopted levels for 16O . . . . . . . . . . . . . . . . . . . . . 136
B.4 List of adopted levels for 20N . . . . . . . . . . . . . . . . . . . . . 137
B.5 Atomic and core masses . . . . . . . . . . . . . . . . . . . . . . . 138
B.6 Calculated Gamow peaks for the reactions of the helium burning
stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
B.7 Calculated Coulomb penetration factors for the reactions of the
helium burning stage . . . . . . . . . . . . . . . . . . . . . . . . . 139
B.8 Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . 139
B.9 Coordinate system for the target and generator system . . . . . . 140
B.10 Base units used in the simulation software . . . . . . . . . . . . . 140
B.11 Ranges for the S-factor extrapolation using AZURE2 . . . . . . . 140
163
164
List of Codes
5.1 IOC shell example – dbLoadRecords . . . . . . . . . . . . . . . . 74
5.2 Example calc record . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3 Example of a Device Support C interface . . . . . . . . . . . . . . 76
5.4 Example of a asynDriver serial Port configuration . . . . . . . . . 78
5.5 Example of a protocol and the corresponding record . . . . . . . . 79
5.6 Example of a record using dev-gpio . . . . . . . . . . . . . . . . . 81
5.7 Example of a thisEPICS.sh file . . . . . . . . . . . . . . . . . . . 84
5.8 Example of the epics.service file . . . . . . . . . . . . . . . . . . . 85
5.9 Example using the udevadmin command . . . . . . . . . . . . . . 86
5.10 Example for a udev rule. . . . . . . . . . . . . . . . . . . . . . . . 86
5.11 Example of the command dbLoadTemplate . . . . . . . . . . . . . 88
5.12 Example of the command dbLoadMultiRecords . . . . . . . . . . 89
5.13 Example of the Profibus StreamDevice bus . . . . . . . . . . . . . 91
5.14 Example of a record using the ADC1248 module . . . . . . . . . . 92
5.15 Example of a calc record converting U_Expo_New format to mbar 95
5.16 Example of the db and proto file for the HMP4040 device . . . . . 97
5.17 Example of some of the SiDet records . . . . . . . . . . . . . . . . 101
C.1 Installation Script EPICS . . . . . . . . . . . . . . . . . . . . . . 141
C.2 Installation Script MXSlowControl . . . . . . . . . . . . . . . . . 143
C.3 Installation Script Module MXFrontend . . . . . . . . . . . . . . . 145
C.4 Bash completion for EPICS . . . . . . . . . . . . . . . . . . . . . 145
C.5 ZSH completion for EPICS . . . . . . . . . . . . . . . . . . . . . . 146
C.6 Header file of the C++ class MUX0 . . . . . . . . . . . . . . . . . . 146
C.7 Source file of the C++ class MUX0 . . . . . . . . . . . . . . . . . . 147
C.8 Header file of the C++ class for the I2C communication . . . . . . 147
C.9 Source file of the C++ class for the I2C communication . . . . . . . 148
C.10 Header file of the C++ class mxlog . . . . . . . . . . . . . . . . . . 150
C.11 Source file of the C++ class mxlog . . . . . . . . . . . . . . . . . . 151
C.12 read_aiSIDET and write_aiSIDET function of MXSiDet . . . . . 152
C.13 Definition of MXSiDet_SetLogLevel . . . . . . . . . . . . . . . . . 154
D.1 TTree structure for the particle and the event . . . . . . . . . . . 157
D.2 Main configuration file of the simulation. . . . . . . . . . . . . . . 157
D.3 Detector setup used in the simulation for Phase 1 . . . . . . . . . 159
D.4 Detector setup used in the simulation for Phase 2 . . . . . . . . . 160
165
166
Acronyms
ADC analog-to-digital converter 90
API Application Programming Interface 90
CA Channel Access 73, 77
CAC CA client 73, 76
CAS CA server 73, 75
cms center of mass system 6
CNO cycle Carbon-Nitrogen-Oxygen cycle 15
CSS Control System Studio 81
DB database 73
DBD database definition file 74
EB External Beam 54
EPICS Experimental Physics and Industrial Control System 72
ER Energy Recovering 54
ERL Extreme Relativistic Limit 37
FLRM five-level R-matrix method 29
GEM Gas Electron Multiplier 61
GPIO general purpose input-output 81
GUI Graphical User Interface 81
HRM hybrid R-matrix method 30
I2C Inter-Integrated Circuit 93
I2C-MUX I2C-Multiplexer 93
IC integrated circuit 96
IOC Input-Output-Controller 73
KM K-matrix method 30
LDM Light Dark Matter 56
MAGIX Mainz Gas Injection Target Experiment 54, 56
MAMBO Milliampere Booster 54
MAMI Mainz Microtron 59, 68
MARC MESA Arc 54
MC Monte Carlo 31, 104
MEEC MESA Enhanced ELBE Cryomodule 54
MELBA MESA Low–energy Beam Apparatus 54
MESA Mainz Energy-Recovering Superconducting Accelerator 54
MXSlowControl MAGIX Slow Control System 71, 82
167
Acronyms
OPI Operator Interface 81
PID particle identification 112
PP chain Proton-Proton chain 13
Profibus Process Field Bus 90
PV Process Variable 75
pyEPICS EPICS Channel Access for Python 80
RM R-matrix method 29
RPi Raspberry Pi 81
SiDet silicon strip detector 93
SPI Serial Peripheral Interface 91
STEAM Small Thermalized Electron Source At Mainz 54
TCP Transmission Control Protocol 77
TLRM three-level R-matrix method 29
ToF Time of Flight 67
TPC Time Projection Chamber 61
UDP User Datagram Protocol 77
ZDT Zero Degree Tagger 65
168
Index
T9, 13 Coulomb penetration factor, 11
αem, see Fine structure constant Coulomb phase shift difference, 22
α-capture reaction, 18 Cross section, 7
δ-function, 39 capture, 8
U_Expo_New, 95 differential, 7
caArchiver, see Archiver elastic, 8
caget, 76 inelastic, 8
camonitor, 76 reaction, 8
caput, 76 stellar nucleosynthesis, 8
dbl, 86 Current conservation
drvAsynIPPortConfigure, 95 electromagnetic, 40
drvAsynSerialPortConfigure, 93 hadronic, 40
makeBaseApp, 100
systemctl, 83 DarkMESA, 56
systemd, 83 Data acquisition, 71
tmux, 84 Database, 73
udevadm, 85 definition, 74
udev, 85 de Broglie wavelength, 8
Device support, 73, 75
Archiver, 86 Driver support, see Device support
Areal particle density, 57
AZURE2, 20, 123 EB mode, 54
Elastic scattering, 6
Bash, 85 Elastic scattering phase shifts, 22
completion, 85 Electro-disintegration, 28
Big Bang nucleosynthesis, 5 Electromagnetic current operator, 41
Breit-Wigner, 8 Electron current, 40
Byte stream, 78 Eletric matrix elements, 51
Energy transfer, 37
Carbon-Nitrogen-Oxygen cycle, 15 Energy-convervation, 39
Catcher, 58 Environment variable, 83
Channel access, 73, 77 EPICS, 72
client, 73, 76 dbLoadRecords, 87
server, 73, 75 dbLoadTemplate, 88
checksum, 94 module, see Community module
CNO cycle, see Carbon-Nitrogen-Oxygen ER mode, 54
cycle
Community module, 78 Felsenkeller laboratory, 26, 125
asynDriver, 78 Final state particle generator, 110, 113
Autosave, 80 Fine structure constant, 9
dev-gpio, 81 Four-momentum, 37
pyEPICS, 80 Fourier transformation, 41
StreamDevive, 79
Control system studio, 81 Gamow peak, 11
operation interface, 81 energy, 11
Cosmic background, 26, 125 width, 11
Coulomb matrix elements, 51 Gas jet target, 58
169
Index
Generalized rosenbluth factors, 45 mxlog, 99
Graphical user interface, 81 documentation, 100
unit test, 100
Hadronic current matrix, 40 module, 100
Hadronic recoil factor, 40 MXSlowControl, 71, 82
Hadronic tensor, 40 ADC1248, 90
Helium burning stage, 16 dbLoadMultiRecords, 88
Histogram, 111, 118 HMP4040, 95
multi-dimensional, 119 MPT200, 93
two-dimensional, 119 MXFrontend, 87
Hoyle state, 18 MXSiDet, see MXSiDet
Hydro-Møller, 54 StreamDevice Profibus, 90
Hydrogen burning stage, 12 tutorial, 90
I2C master, 96 Natural system of units, 37
I2C multiplexer, 96 Nuclear reaction, 5
I/O hardware, 75 Nuclear response functions, 43, 131
Importance sampling, 106 transverse-longitudinal interference, 43
Inelastic scattering, 6 longitudinal, 43
Inital channel, 6 transverse, 43
Initial state particle generator, 109, 112 transverse-transverse interference, 43
Input-Output-Controller, 73
shell, 73 P2 experiment, 55
Invariant mass, 46 Phase shift difference, see Coulomb phase
Invariant matrix element, 40 shift difference
IOC, see Input-Output-Controller Photo-disintegration, 27
Poisson distribution, 120
Laval nozzle, 58 PPI-III chain, see Proton-Proton chain
Legendre polynomial, 7, 50 Process variable, 75
associated, 50 Protocol, 79
Leptonic tensor, 40 file, 79
Lorentz boost, 47 Proton-Proton chain, 13
Lorentz transformation, 46 Python, 80
Luminosity, 67 script, 80
LUNA-MV, 25, 125
Radiative capture, 6
MAGIX, 56 Random number, 105
Phase 0, 68 pseudo-random, 105
Phase 1, 68, 122 quasi-random, 105
Phase 2, 69, 122 Reaction channel, 6
MAMBO, 54 Reaction rate, 11
MARC, 54 distribution, 11
Maxwell-Boltzmann distribution, 9 Record, 73
Mean value, 104 field, 74
MEEC, 54 type, 73
MELBA, 54 Reduced mass, 9
MESA, 54 Resonance, 8
metastable, 17 strength, 8
Metric tensor, 38 Rodrigues’ formula, 51
Monte Carlo integration, 104 ROOT, 111
Multi-pole operator phase, 51 TTree, 120
MXSiDet, 92
frontend-board, 96 S-factor, 9
HDMI distribution board, 96 cascade transition, 21
library, 96 E1 contribution, 21, 29, 123
AD5693, 98 E2 contribution, 21, 29, 123
ADS122C04, 98 ground state transition, 21
MXP9808, 98 SiDet, 93
PCA9548A, 99 Silicon strip detector, 63
RPiI2C, 97 test stand, 93
SiDet, 99 Simulation, 103
170
Index
acceptance, 111, 116 Three-momentum transfer, 37
coincidence, 118 Time Projection Chamber, 61
set, 118 Transition charge density, 41
silicon strip detector, 117 Transition current density, 41
spectrometer, 117 Transition matrix element, 41
zero degree tagger, 117 Transmission control protocol, 77
analysis, 118 Trigger veto system, 62
count rate, 119 Triple-α reaction, 17
cross section, 108, 111 User datagram protocol, 77
event generator, 109
physics model, 110, 115 Variance, 104
Slow control, 71 Virtual photon, 37
Sobol sequence, 105
Sommerfeld number, 9 Weak mixing angle, 55Weinberg angle, 55
Spherical unit vectors, 41
STEAM, 54 Zero degree tagger, 65
Stellar nucleosynthesis, 5 ZSH, see Bash
171
172
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