Simulation studies for an X-ray polarization satellite Master of Science Thesis

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Simulation studies for an X-ray polarization satellite

Master of Science Thesis

HADRIEN VERBOIS

Stockholm December 2012

Particle and Astroparticle Physics

School of Engineering Science

KTH Royal Institute of Technology

TRITA-FYS 2012:63

ISSN 0280-316X

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Abstract

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5 Detector characterization for a realistic GRB spectrum 77 5.1 GRB spectrum . . . 77 5.1.1 A realistic GRB spectrum . . . 77 5.1.2 Geant4 implementation . . . 78 5.2 Off-axis Sources . . . 80 5.2.1 Renormalization . . . 80 5.2.2 Modulation curve . . . 81 5.2.3 Efficiency . . . 83 5.3 Detector characterization . . . 85

5.3.1 Minimum detectable polarization . . . 85

5.3.2 Detector response dependence on the off-axis angle . . . 85

5.3.3 GRB 100826A . . . 88

6 Conclusion and outlook 89 6.1 Conclusion . . . 89

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Chapter 1

Introduction

1.1 Polarization in physics

1.1.1 Polarization

Polarization of light is defined as the orientation of the electric field Exyin the plane perpendicular

to the wave direction of propagation. If Exy= Ex(t) + Ey(t), then the wave polarization is said to

be: • Circular ifEx(t) Ex(0) 2 +Ey(t) Ey(0) 2 = 1. • Linear if |Ex(t)| and |Ey(t)| are constant.

• Elliptic if it is a combination of the two previous cases. • Unpolarized if Exand Ey are randomly distributed.

Light polarization arises from quantum behavior of photons. Individual photons are in a certain polarization state. This state can be expressed within a base of two states: horizontal (<→ |) and vertical (<↑ |). The polarization of a beam depends on the polarization states of the photons that compose it. For instance a beam of photons in the pure horizontal state is horizontally polarized. A combination of an equal number of photons in horizontal and vertical states forms unpolarized light, whereas a bulk of photons in the state (<→ |+ <↑ |) gives circularly polarized light.

A beam is said to be partially polarized if it is a superposition of polarized and unpolarized light. One uses the degree of polarization Π to describe the ratio between those two components:

Π = Ipolarized light

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For example, an unpolarized beam has a polarization degree of 0% whereas a totally polarized beam has a polarization degree of 100%.

1.1.2 Polarized X-ray emission mechanisms

Magneto-bremsstrahlung radiation

Magneto-bremsstrahlung emissions are due to mechanisms involving a charged particle moving with a velocity v in a constant magnetic field B. The force acting on the particle is given by:

F = Ze

c (v × B) where Ze is the particle charge, and c the light speed.

It is perpendicular to both B and v. Thus, if v and B are orthogonal, the particle will have a circular path. If the angle χ = (v, B) (the pitch angle) is not 90◦, then the particle has a helical path. In both cases the particle has a constant acceleration and rotates around the magnetic field line. It is straightforward to compute the so called gyro frequency νr:

νr=

ZeB 2πγmcc

. (1.1.1)

Using the Lamor formula, the energy rate of the radiation can be computed: dE dt = σt 4π v c 2 cγ2B2sin2χ (1.1.2) where σt= 8π(Ze)4/3m2cc

2_{is the Thomson cross section, γ =} _√ 1

1−v2_/c2 and mcthe charged particle

mass. The radiation power is proportional to m−2_c . That is why in most of the cases, the charged particles emitting Magneto-bremsstrahlung radiations are electrons.

There are two particular cases of magneto-bremsstrahlung: • non relativistic: cyclotron radiation;

• ultra relativistic: synchrotron radiation.

Cyclotron Radiation: In the non relativistic limit, v << c and therefore γ = 1. Hence the radiation rate becomes:

dE dt = σt 4π v c 2 cB2sin2χ (1.1.3)

The power radiated varies as sin2_{θ with respect to the acceleration vector a, as can be seen in}

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Figure 1.1: Polar diagram showing the dipole radiation emitted by an accelerated electron (3).

Polarization

The polarization vector lies in the plane (a, p), where p is the photon momentum (3). Thus:

• if the line of sight is perpendicular to B, then the light will be 100% linearly polarized; • if the line of sight is parallel to B, then the light will be 100% circularly polarized;

• if the line of sight is neither perpendicular nor parallel, the light will be elliptically polarized.

Synchrotron Radiation: In the rest frame of the electron, the radiation power follows the same pattern as for the cyclotron emission. But in the observer frame, it is modified by relativistic beaming, as shown in Figure 1.2.

Figure 1.2: The dipole radiation emitted by a relativistic accelerated electron as transformed into the observers frame of reference (3).

Polarization

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electron, the more the energy is radiated in v, i.e. in the limit case, only an observer with a line of sight perpendicular to B will see any radiation. And as for the cyclotron emission, the X-rays are linearly polarized in this case.

Bremsstrahlung radiation

Also called free-free emission, it is caused by the acceleration of a charged particle in the electro-static field of an ion. As for the case of magneto-bremsstrahlung emission the energy loss due to bremsstrahlung radiation is more important for electrons than for heavier particles because they experience greater acceleration due to their smaller mass.

The maximum energy of a photon emitted from bremsstrahlung radiation, for any two charged particles, is given by (10),

Emax=

mzE1

m2+ m1+ E1− M cosθ

(1.1.4) where E1 , M , and m1 are the kinetic energy, momentum, and rest mass of the incident particle

respectively, m2 is the rest mass of the target particle and θ is the emission angle measured from

the incident direction of the incident particle. Polarization

According to (11), the degree of linear polarization of the emission from electron-proton bremsstrahlung radiation is given by:

Π = mec 2_{< zD} 0− 2m3ec6 E2_∆ 0− mec2E∆0+ 2m3ec6 (1.1.5) where E is the energy of the emitted photon, ∆0= Ee− M cosθ, and Ee is the initial total energy

of the electron (rest mass and kinetic energy).

A plot of the variation of the degree of linear polarization with the emitted photon energy for an initial electron energy of Ee= 6mec2 is shown in Figure 1.3. The degree of linear polarization

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Figure 1.3: The variation of the maximum energy (in units of mec2) of emitted photons from

electron-proton (dashed lines) and proton-electron (solid lines) bremsstrahlung with the direction of emission for various incident particle energies. The lines correspond to equivalent electron and proton velocities (10).

Compton scattering

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Figure 1.4: Schematic view of a Compton scattering: Eγ and E’γ are the electron initial and final

energy respectively; θ is the polar scattering angle.

Polarization

Compton and inverse Compton scattering can both produce polarized photons from an initially unpolarized flux. Scattering is also responsible for the depolarization of polarized beams, and thus an originally polarized flux undergoing multiple scattering will have a substantially reduced degree of polarization.

The degree of polarization observed from Compton and inverse Compton scattering is dependent on the vectors of the incident photons being aligned. For an isotropic distribution of incident photons the scattered beam will be composed of polarized components from all incident directions, these components will cancel leaving a completely non-polarized beam.

1.2 Polarimetry

1.2.1 Physical processes

Photoelectric effect

In the photoelectric effect, a photon is absorbed by an atom. The photon energy is transfered to an electron. If this electron binding energy is smaller than this transmitted energy, this electron is emitted with a kinetic energy:

Ek= hν − Ebind (1.2.1)

where h is the Planck constant, ν the photon frequency and Ebindthe binding energy of the electron.

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where r0 is the classical electron radius, Z is the atomic number of the absorbing material, α is

the fine-structure constant, me is the mass of the electron, c is the speed of light, θ is the polar

emission angle of the electron, φ is the azimuthal angle of emission from the polarization plane and β = v/c with v being the final velocity of the emitted electron.

Due to the cos2φ dependence, the azimuthal electron emission angle will be modulated by the polarization of the incoming photon. Thus, by measuring the distribution of the electron azimuthal angle, one can determine the polarization of the source.

Compton scattering

The Compton scattering differential cross section is given by the Klein-Nishina formula: dσ dΩ= r 2 0 E02 E2( E0 E + E E0 − 2sin 2_θcos2_φ) _(1.2.3)

where r0 is the classic electron radius, θ is the polar scattering angle and φ is the azimuthal

scattering angle with respect to the initial polarization vector. E and E’ are the initial and final photon energies respectively. The relation between E and E’ is given by formula 1.2.4.

E0= E

1 + E

mec2(1 − cosθ)

(1.2.4)

Due to the cosφ2 _{dependence, the azimuthal electron emission angle will be modulated by the}

polarization of the incoming photon.

Pair production

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Figure 1.5: Three dimensions geometry of pair production.

The process tends to be coplanar, and then according to (2), its differential cross section is given by:

dσ

dΩ= A(1 − cos

2_ψ), _(1.2.5)

where A ≈ 0.8 is the asymmetric ratio L ≈ 0.2 is the degree of asymmetry in the distribution. As in the case of Compton scattering the distribution will show a modulation due to the cos2_ψ

dependence.

Energy range

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Figure 1.6: Mass attenuation coefficient cross section for photon interactions in water (6). The contribution of photo-absorption, Compton scattering and pair production can bee seen.

1.2.2 Modulation factor

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Figure 1.7: The counting rate dependence in θ can be fitted by the sinusoidal function N (θ) = Asinφ + B. φ is the angle between the outgoing photon direction and a reference direction in the plane perpendicular to the incoming beam direction.

In order to quantify the degree of polarization, one can define the modulation factor µ as: µ = Cmax− Cmin

Cmax+ Cmin

, (1.2.6)

where Cmax and Cmin are the maximum and minimum counting rate respectively, as shown in

Figure 1.7. The modulation factor for a 100% polarized beam is µ100. This value depends on the

detector, it can be obtained through simulations or through measurements where the polarization degree is known, e.g. at accelerators.

Using µ100, one gets the polarization degree Π of a photon flux as:

Π = µ µ100

. (1.2.7)

Another way to define the modulation factor, more directly linked to the detector itself is: µ = N⊥− Nk

N⊥− Nk

, (1.2.8)

where N⊥ and Nk are the count rate in orthogonal detectors in the XY plane. In particular,

for point scattering and point detection, the expected count rate is given by the differential cross section. Thus,

µ = dσ(η = 90) − dσ(η = 0)

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For Compton scattering, using Klein Nishina formula, one gets: µ = sin

2_θ

−1_{+ − sin}2_θ. (1.2.10)

Figure 1.8 displays the modulation factor with respect to θ for different energies. It shows that for increasing photon energy, the maximum modulation factor is achieved at lower θ. Photons scattering forward or backward do not carry any information about the polarization.

Figure 1.8: Modulation factor dependence to the scattering angle for point scattering and detection for different incident photon energies.

The designs of the polarimeters considered in this work are such that one actually integrates the cross section over several θ:

µ =σ(η = 90) − σ(η = 0) σ(η = 90) + σ(η = 0), (1.2.11) where σ(η0) = Z θ2 θ1 dσ(η0).

For instance, for a 100 keV photon beam, if one considers the photons scattered between θ1= π/8

and θ2= π − π/8, the modulation factor is µ = 0.47.

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theoretical modulation factor is not as easy as in the previous cases, since one cannot directly integrate the cross section between the allowed angles, but must take in account the geometrical modulation brought by the cylinder. For simplicity, the calculation is restricted to the case where the photons are only irradiating the center of the cylinder.

Figure 1.9: Toy model for a polarimeter. Photons escaping through the top and the bottom are not detected.

Depending on where on the z axis the Compton scattering occurs, only photon with certain θ angles will be scattered towards the side of the detector, and thereby be detected:

• for θ < θmin and θ > θmax= π − θmin, the photon will never be detected;

• for θ = π/2, it will detected for all z.

If one assume that the probability of having a Compton scattering is not z-dependent, one can construct the detection probability density shown in Figure 1.10.

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Modulating the cross section by this detection probability density, one gets, for the toy model and for 100% polarized photons at 100 keV a modulation factor µ = 0.59.

Figure 1.11: Modulation factor dependence on the energy for different integration boundaries. Figure 1.11 synthesizes this section by showing different modulation factor dependences on the energy. It will be very hard to anticipate a real detector modulation factor. This is why simulations are needed. Nevertheless, it should be noticed that the modulation factor decreases with the energy.

1.3 Science motivation

1.3.1 Gamma-ray burst

Gamma ray bursts (GRB) are highly energetic flashes of X- and gamma-rays. They are the most powerful events ever observed in space. They were first detected in 1967 by the US Vela satellites, which were dedicated to the observation of gamma rays from nuclear weapons tested in space, but the discovery was only made public in 1973. Several theories were proposed to explain such bursts, many involving nearby sources in the Milky Way. In 1993, the BATSE experiment showed that GRBs are isotropically distributed, which ruled out most of the galactic models.

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last less than 2 s, and the long GRBs, that last more than 2 s.

GRBs are followed by fading emission at longer wavelengths (from X-ray to radio), called after-glow (AG). Their discovery enabled the identification of the host galaxies and of the corresponding redshift. The first successful measurement was made in 1997: (12) measured a z=0.835 for GRB 970508. Together with many measurements realized later on, this proves that the GRBs take place in extremely distant galaxies. In 2005, the mean redshift for one particular GRB was estimated at 2.8 by (13). With z=9.4, GRB 090429B is the oldest ever detected (it occurred when the universe was 520 million years old).

GRB is a cutting edge topic. There exist many different models for the GRBs’ emission mech-anisms. However, most of them agree on certain features. First of all, it is widely accepted that GRBs involve a relativistic motion with a high Lorentz factor Γ. This is motivated by the compact-ness argument, and has been validated by several observations. For example, (15) suggests that the afterglow of GRB 990123 indicates a ultra relativistic motion with Γ ≈ 100. Secondly, GRB emissions are believed to be released in a jet. The observational motivation for this resides in the monochromatic break that occurs in many afterglow light-curves. In most models, the energy of the GRB and the afterglow is provided by dissipation of the relativistic flow. But the mechanisms providing this dissipation differ from one model to another. Currently, there are two main theories: the Fireball models (FB) and the Cannonball models (CB).

The first FB models were suggested by (7) in 1986 and have evolved since then. They assume that GRBs are emitted from highly relativistic conical fireballs produced in hypernovae. The GRB pulses are assumed to be produced by synchrotron emission from internal shocks, and the AG is assumed to be synchrotron radiation from external shocks. Internal shocks occur when highly relativistic conical shells collide with each other whereas external shocks occur when a shell collides with the interstellar medium. Figure 1.12 illustrates the FB model.

Figure 1.12: Fire Ball (FB) model for GRB (24).

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factor Γ in opposite directions, along the rotation axis. The X- and gamma-rays of a GRB single pulse are believed to be produced when the CB goes through the SN glory (the light scattered by the SN ejecta). The photons from this latter then undergo inverse Compton scattering off the electrons enclosed in the CB. The CB expands rapidly and the AG is emitted first by thermal bremsstrahlung and then by synchrotron radiations when the CB scatters off the interstellar medium (ISM). Figure 1.13 shows the main assumptions of the FB and CB models of long duration GRBs.

Figure 1.13: Main assumption of the FB and CB models of long duration GRBs (24).

Most of CB and FB models are in agreement with the observations of the spectrum or light curve of GRBs. Polarimetric observations on the other hand could help to discriminate between those models. Originally, polarization was expected to be very small for FB models, and high for CB models (18). But a very large GRB polarization was reported for GRB 021206 (19). Although the results were criticized (20), it prompted FB model papers on GRB polarization, showing that under certain circumstances, and in particular under certain viewing angles, FB may also produce a large linear polarization (21; 22; 23).

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(a) SO (b) SR

(c) CD

Figure 1.14: Linear polarization degrees in the 60 to 500 keV band as a function of q = θv/θj ,

where θv is the viewing angle of the observer and θj is the jet opening angle, for several values of

yj = (Γθj)2, calculated in the different models (16). Γ is the jet lorentz factor. SO, SR and CD

refer to the ordered synchrotron model, the random magnetic field synchrotron model and to the Compton drag model respectively.

1.3.2 Previous missions and results

The first attempts to measure GRB polarization have used non-dedicated instruments. The first claim of detected polarization (19) was already mentioned; it was realized using the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI), which was dedicated to observing solar flares. The INTEGRAL satellite was also used to measure the polarization of GRB 041219A (26; 27; 28). In both cases, the statistics were very low and the results could not be confirmed by subsequent analysis.

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There are also a few proposed polarization missions optimized for GRBs. The POLAR instru-ment, proposed for the Chinese Space Station (33), and TSUBAME, a small satellite mission (34) are both in the development phase. GRAPE is a balloon borne mission, expected to do a long duration (10 days) flight at the South Pole in 2017. Although it first aims at solar flares, it has the potential to observe GRB polarization (35).

1.3.3 SPHiNX

Polarimetry of GRBs is a research field in its infancy, and there is a need for different observations. The proposed Segmented Polarimeter for High eNergy X-ray (SPHiNX) mission is intended as a small satellite payload dedicated to the measurement of X-ray polarization of transient events such as GRBs. Therefore, the SPHiNX instrument will have a large field of view. Its exact energy range must be investigated, but, since the SPHiNX instrument is to be a Compton polarimeter, it will lay in the 20 to 500 keV window.

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(a) GAP-like design (b) GRAPE-like design

Figure 1.15: Schematic overview of the two types of polarimeter design.

In the first design, which will be referred to as the GAP-like design, each polarimeter unit comprises a central plastic scintillator surrounded by a number of high atomic number inorganic scintillators (CsI or BGO). Each scintillator piece is coupled to a separated PMT. A possible design is shown in Figure 1.15(a). In the ”GRAPE-like” design, each polarimeter unit comprises one single multi-anode photomultiplier tube (PMT). Each anode segment is coupled to a scintillator bar of length 50 mm in order to restrict the polar scattering angle. The 28 outer bars are CsI or BGO while the inner 36 bars are plastic. The design is shown in Figure 1.15(b).

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Chapter 2

Simulation

2.1 Introduction to Geant4

2.1.1 Geant4 use

Geant4 (GEometry ANd Tracking) is a platform for the Monte Carlo-based simulation of particle-matter interaction. It is the latest version of the Geant series, developed by CERN since 1974. It is the first version based on C++.

Geant4 provides tools for all area of detector simulation: geometry, physics, visualization, event management, user interface etc... The definition of the parameters is done either by implementing subclasses or through the use of a macro.

In our case, the main aspects that need to be considered are: 1. Detector geometry.

2. Definition of the physics involved in the simulation. 3. Selection of simulation output.

4. Definition of the particle flux irradiating the detector.

2.1.2 Short review of Geant4 mechanism

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particles if some are created in a step. The details of the algorithms used in those objects can be found in (8).

Geant4 provides the user with several functions, that are called at certain times of the tracking. For example, the function UserSteppingAction is called each time a step ends, and it has access to all the attributes of the step and its corresponding track.

Another way to access information about the tracking is to use so called active detectors. If those are implemented, at the end of each event (an event starts when one primary particle is produced and stops when this primary and all the secondaries it created have been terminated (null kinetic energy or particle leaving the geometrical boundaries)), the user will have access to all the steps that took place in the detector. Moreover, the user can apply filters and thereby select the interactions that are of interest.

An important feature in the Geant4 tracking mechanism is that all the particles involved are treated one after another. The tracking manager treats one track until it terminates, and only then it starts taking care of the secondaries of this track. This implies that there is no interaction between particles in Geant4, but only between particle and material. It also means that the order in which Geant4 handles events is not the actual time-order.

2.2 Detector geometry and physics involved

2.2.1 Compton scattering in the scintillators

Choice of materials

The Geant4 conception of material is based on the fact that in nature, every material (i.e. mixture, chemical compound, etc) is made of elements and that each element can have several isotopes. Therefore, three main classes are involved in material description: G4Material, G4Element and G4Isotope. The latter will be omitted in what follows since it is not used in this thesis.

The G4Element class describes the properties of the atom such as the atomic number, the number of nucleons, the atomic mass, the number of shells and their energy. There are two ways to create a G4Element: either by specifying its atomic-number and nucleon-number, or by giving its isotopes. Geant4 automatically computes the electronic structure of the atom.

The G4Material class describes the macroscopic properties of matter such as density, temperature and pressure. There are many ways to create a G4Material, as a mixture of several elements for example.

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Processes and particles

All the physical properties of a simulation must be defined in the physics list. There are two mandatory steps: the definition of the particles involved in the simulation and the definition of the processes associated to those particles.

Particle definition: Geant4 allows the construction of new particles. It also furnishes a list of particles ready to use, that will be sufficient for this thesis.

Process definition: The physical processes define how particles interact with matter. In Geant4, all the processes that are to be considered for each type of particle must be explicitly defined. Seven major categories of processes are provided by Geant4:

1. electromagnetic, 2. hadronic, 3. decay, 4. photolepton-hadron, 5. optical, 6. parameterization, 7. transportation.

Each process may have different models and cross sections, which specify for instance the energy range or the response to the polarization (for electromagnetic processes e.g). In addition to those

Figure 2.1: Particles, processes, cross-sections and models

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production threshold of a particle, or set a tracking cut (an energy under which the tracking of the particle is terminated).

There are a lot of processes already implemented, but some are not included in the basic Geant4 package and must be downloaded separately. It is also possible for the user to define new process(es).

Physics List: The high flexibility of Geant4 regarding the definition of the physics list requires caution: once a physics list has been built and compiled, it must be validated. The Geant4 collab-oration does not define a procedure to validate a physics list, it is the user’s responsibility to design a series of tests for this purpose.

In certain cases, it is possible to avoid this step by using one of the ready-to-use physics lists available in the Geant4 package. Unfortunately there are not many of those lists and it is sometimes complicated to find a complete description.

2.2.2 Scintillation and light collection

Principle of the detection

Certain materials emit a small flash of light when struck by radiation, this is the process of scin-tillation. The light emitted is typically in the optical range and for scintillator materials, there can be more than 10 photons emitted per keV. When the scintillator material is coupled to a photomultiplier, this flash of light can be converted to an electrical pulse and analyzed.

Implementation with Geant4

Optical processes in Geant4: For the treatment of low energy photons, Geant4 has introduced the concept of optical photons. They are treated as particles just as electrons or photons of higher energy. It introduces a discontinuity in the description of the light: for high energy, one uses the G4photon particles, and for low energy (visible light), one uses G4OpticalPhoton. In our case the question of where the limit is to be set is not an issue: the ”optical photons” intervene only in the scintillation process, whereas the X-ray and gamma-ray are represented by G4photon.

Geant4 provides the following processes for optical photons: • Optical photons production:

– Cerenkov process. – Scintillation process. – Transition radiation. • Other processes:

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– Bulk absorption/optical absorption. – Rayleigh scattering.

– Wavelength shifting.

In order to implement optical processes, the properties of each process must be specified, but also the properties of the materials where optical processes are expected to take place.

Scintillation process: In particular, the scintillation process settings are:

• The spectrum of the emitted photons, possibly for a fast and a slow component. • The time constant for the fast and the slow component.

• The ratio fast/slow component.

• The light yield, as a function of the energy.

• The energy resolution scale (1 gives a Poisson law, 0 a ”perfect” resolution).

Light path: The path followed by a scintillation photons depends on the material properties and on the processes defined in the physics list. In this thesis, the boundary processes and the optical absorption will be considered.

• Material bulk properties:

Considering the processes of interest in this work, the following material properties need to be defined:

– The refractive index as a function of the energy. – The absorption length as a function of the energy.

• Boundaries: The description of the boundaries between materials and components can be done very thoroughly in Geant4. It uses the concept of surfaces, and involves two classes:

– The G4LogicalSurface defines the physical or logical volume bordering the surface. – The G4OpticalSurface keeps informations about the physical properties of the surface

itself (e.g. polished, rough, etc.).

This latter class can use two different simulation models: GLISUR-model and the UNIFIED-model. More details about those two models can be found in (4).

If the user does not define any specific optical surface but still calls the boundaries process, Geant4 will handle the change of medium using Snell’s law.

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2.3 Electronics & treatment of the events

Another task of the detector is the reception and the treatment of the signal emitted by the PMTs. This signal is first read by an analog circuitry and then digitized so that it can be analyzed by an embedded circuitry, a microprocessor or even by a ground station, provided that the detector has a communication system. Geant4 is not meant to simulate this part, and ROOT (31) will be used instead.

When a photon interacts with a scintillator, it leaves a footprint, the deposited energy. This is called a hit. The set of all the hits caused by one photon is called an event. Because of the velocity of the photon, the detector cannot distinguish the order of the hits in one event. For the same reason, if several interactions happen in the same scintillator, for the same photon, the circuitry cannot distinguish them, and see them as one hit. Therefore, the only thing that can be read for each event is whether or not each scintillator has had a hit and what the energy deposit of this hit is.

2.3.1 Storage of the hits

Geant4 makes it possible for the user to study a single interaction in detail. In order to store the Geant4 data, an array where one bin corresponds to one scintillator (or one PMT) is used. Each time Geant4 emits a new photon, all the bin values are set to zero. Then, for each interaction, the energy deposit is added to the bin corresponding to the scintillator where the interaction happens. By that means, for each incoming photon, a map of the deposited energy in the detector is obtained. An advantage of this method is that it provides the same kind of data to the analysis part that would be available in reality.

2.3.2 Energy thresholds

In a real detector, there is noise from the electronics, form thermal emission etc. Therefore, in order not to read out channels with no activity, only channel which deposit energy above a certain level are recorded. In our simulations there is no electronic noise, but it must be taken in account that in a real detector, this threshold setting will also lead to the elimination of some valid events.

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2.3.3 Event selection

GAP like: 1. 1-hit events

A 1-hit event can occur in several ways. The most likely are the following: (a) The photon was absorbed without scattering.

(b) The photon underwent a Compton scattering and then escaped the detector. (c) The photon underwent two interactions, but only one was recorded.

(d) Two hits occurred in the same scintillator and were thus detected as one.

Even if there is a Compton scattering, the azimuthal angle cannot be known, so these events are not taken into account.

2. 2-hit events

These events can be CsI-CsI or Plastic-CsI events. They are always taken into account. 3. 3-hit events

The two possible patterns for a 3-hit event are two Compton scatterings followed by a photo-absorption and three Compton scatterings. The order in which the hits occurred is unknown, but in the first case some assumptions can be made in order to discriminate two hits, and take the event into account:

(a) The hit with the highest energy deposit corresponds to the photo-absorption, and is kept.

(b) among the two remaining hits, the most energetic is taken into account, for according to equation 1.2.4, the lower the energy deposit, the less scatters the photon.

In the second case on the other hand, it is not possible to discriminate two hits, and the event must be rejected.

GRAPE like: 1. 1-hit events

For the same reasons as in the GAP-like model, these events are not taken into account. 2. 2-hit events

There are three types of 2-hit event: (a) Plastic-plastic

(b) Plastic-CsI (c) CsI-CsI

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3. 3-hit events

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Chapter 3

Study of Geant4

In a Compton polarimeter, X-rays are expected to scatter in one scintillator and be subsequently photo-absorbed in another scintillator. In both Compton scattering and photo-absorption, a sig-nificant percentage of the photon energy is deposited in the material where the interaction takes place. Scintillator materials convert this deposited energy into optical photons that can be detected by a photo-multiplier tube (PMT).

In this chapter the basic processes and materials involved will be investigated. Geant4 implemen-tation of Compton scattering and photo-absorption in a scintillator will be first considered. Then, the scintillation and the light collection will be analyzed.

3.1 X-ray interactions in Geant4

3.1.1 Materials

The photo-electric cross section is proportional to the atomic number to the power of five. Therefore, the scintillators are arranged such that a photon Compton scatters from a low atomic number scintillator to a high atomic number scintillator. Plastic scintillators are usually used for Compton scattering, whereas CsI or BGO (Bismuth Germanium Oxide) are favored for photo-absorption.

The Geant4 material database contains a vinyl-toluene plastic scintillator that will be used in this thesis.

Neither CsI nor BGO is available in the Geant4 database. But they can both be defined as a composite material. For the sake of simplicity, CsI (Cesium + Iodine) is chosen over BGO (Bi4Ge3O12).

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are not supported by Geant4. Therefore, they are set to arbitrary low values: (8) recommends 10−19 Pa and 0.1 Kelvin.

3.1.2 Processes

Compton scattering and photo-absorption

Compton scattering and photo-absorption are the two fundamental mechanisms in a Compton po-larimeter (concerning X- and γ-rays). Other secondary reactions can take place in the scintillators, but they will not be considered in this thesis.

The Geant4 implementation of those two mechanisms requires the use of three G4Processes: G4ComptonScattering, G4PhotoelectricEffect and G4eIonisation. The two first ones concern pho-tons and the last one electrons. For each process (Compton scattering and photo-absorption), two properties have to be checked: the cross section and the deposited energy. This requires two simulations.

Figure 3.1: A beam of photons with a flat energy spectrum between 10 and 100 keV irradiates a 10 cm × 2 cm cylinder of plastic scintillator.

In the first one a beam of photons with a flat energy spectrum between 10 and 100 keV is directed towards a 10 cm × 2 cm cylinder of plastic scintillator, as can be seen in Figure 3.1. Accessing Geant4 tracking information, each time a photon Compton scatters or is photo-absorbed, its initial energy is recorded. Figure 3.2 shows the number of interactions with respect to the incoming photon energy, for each process. As required, it matches the Compton scattering and photo-absorption cross sections, given in Figure 1.6.

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The abrupt cut in Figure 3.3(a) corresponds to the Compton edge. Its theoretical value is given by equation 3.1.1 and is in accordance with the value read on the graph ≈ 8 keV.

Ee=

2E2 mec2+ 2E

= 8.1 keV (3.1.1)

For the photo-absorption, the deposited energy is the same for all reactions: Ephoto= Ei−288 eV,

where Ei = 50 keV is the incident energy. The binding energy of the outer shell electron of the

carbon is 288.23 eV. Therefore, the peak position in Figure 3.3(b) agrees with equation 1.2.1.

(a) Compton scattering (b) Photo-absorption

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(a) Compton scattering (b) Photo-absorption

Figure 3.3: Deposited energy for 50 keV incoming X-rays. One million photons have been used per simulation.

Polarized Compton scattering

The default process for Compton scattering in Geant4 does not take the polarization into ac-count. A dedicated model must be attached to the G4Compton scattering: the G4Livermore-PolarizedCompton model. In order to test this model, the modulation curve must be checked.

For each Compton scattering, the azimuthal angle is recorded (angle between the momentum of the outgoing photon and a reference vector in the XY plane). Figure 3.4 shows the resulting modulation curve. As expected, there is a sinusoidal modulation. But there is no theoretical reference to verify that the modulation factor (µ100≈ 0.48) is reasonable.

However, Figure 1.8 gives the theoretical modulation factor for point scatter and detection. At 50 keV, the maximum is reached for a scattering angle θ ≈ π/2: µπ/2 ≈ 0.95. The previous set-up

is reproduced, but the azimuthal angle is recorded only when the scattering angle is close to π/2 (between 89◦ and 91◦). The resulting modulation curve is shown in Figure 3.5. The modulation factor is then µexp,π/2≈ 0.9. This is slightly lower than the theoretical value. It is expected, since

this latter is computed for θ = 90◦, whereas µexp is simulated for 89◦< θ < 91◦, and according to

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Figure 3.4: Number of interactions with respect to the azimuthal angle, when a 100% polarized on-axis beam of 50 keV photons is aimed at a the cylinder of plastic scintillator shown in Figure 3.1.

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3.1.3 Active detector

The objective of this section is to test the active detector class of Geant4. A very simple detector is simulated. Its design is shown in Figure 3.6. The materials and processes used are those that have been tested in the previous sections. The CsI scintillators are defined as active detectors. A filter is implemented, so that only photo-absorption interactions are recorded.

Figure 3.6: Design of a simple polarimeter. The blue material is plastic scintillator and the red one is CsI. The rest of the chamber is filled with vacuum. The CsI scintillators are defined as active detectors, and a filter is implemented so that only photo-absorptions trigger a hit.

A monochromatic beam of 100 keV photons is directed at the detector, on axis. They are all polarized in the y direction. The number of hits in each CsI segment is the following:

CsI detector1: 67072 counts CsI detector2: 66428 counts CsI detector3: 140845 counts CsI detector4: 140605 counts µexp= 0.41.

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CsI detector1: 103591 counts CsI detector2: 103736 counts CsI detector3: 104153 counts CsI detector4: 103808 counts µexp= 0.

This polarimeter design is not interesting for a real measurement, since e.g its performance is too much depending on the polarization angle of the observed flux.

3.2 PMT and scintillation process

The scintillation process and the light collection mechanism are fundamental in a detector, for they convert the deposited energy to a readable electric signal. In Geant4 simulations, this function can be divided in three parts: the scintillation, the propagation of the scintillation photons through the material, and the detection of such photons by a PMT.

3.2.1 Materials

As explained in chapter 2, in Geant4 the optical properties of a material are defined independently of the material itself. It it therefore the user’s responsibility to implement a consistent description. Saint-Gobain Crystals (29) provides an extensive description of its plastic scintillators. In par-ticular, the BC-408 is a standard plastic scintillator made of poly-vinyl-toluene. Although it might not be the optimum material for SPHiNX, it is consistent with the plastic scintillator already implemented in the simulations (vinyl-toluene).

Referring to (30), the optical properties of the plastic scintillator are defined as follow:

• Geant4 provides a fast and a slow component for the scintillation process. Only the fast one is activated.

• The energy resolution follows a Poisson distribution.

• The spectrum of the scintillation photons is given manually. Eight points are taken from Figure 3.7.

• Light yield: in Geant4, it can be energy dependent. However, Saint-Gobain Crystals only provides a single value: 13600 photons/MeV.

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Figure 3.7: Scintillation emission spectra for BC-408 (30)

3.2.2 Scintillation process

The scintillation process in Geant4 considerably slows down the simulations, since each time an X-ray interacts with the material, a large number of photons is created. Therefore, it is reasonable to limit the number of incoming photons. A ten thousand photon beam for example will be processed for several hours.

A beam of one thousand 50 keV mono-energetic photons is directed at the plastic scintillator cylinder shown in Figure 3.1. For each interaction in the scintillator, the sum of all scintillation photons energies is recorded.

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Figure 3.8: Number of reactions in the scintillator in function of the energy deposited in the scintillator by scintillation.

3.2.3 Detection of the scintillation photons

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Figure 3.9: Model coupling of a plastic scintillator and a PMT. The plastic scintillator (in black) is a cylinder with a radius of 1 cm and a height of 10 cm . The PMT photo-cathode (in red) is a cylinder with a radius of 9 mm and a height of 2 mm and is surrounded by a 1 mm thick glass (in black).

A PMT is modeled with two parts: a 1.5 mm thick plate of quartz (SiO2) and a 1 mm thick alu-minum plate (the photo-cathode). This model is inspired by (42). A cylinder of plastic scintillator is coupled to this PMT as shown in Figure 3.9. Two simulations are run. In the first one, no special interface between the scintillator and the vacuum is defined. It means that any photon reaching a boundary will escape. In the second one, the plastic scintillator is coated with a 100% reflective mirror. In both cases, a beam of ten thousand 70 keV photons is directed at this detector, and the deposited energy in the PMT is recorded.

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(a) With mirror: 99.5% of the interactions are detected (b) Without mirror: 98% of the interactions are detected

Figure 3.10: A beam of mono-energetic 70 keV photons is fired at the detector. The number of reactions as a function of the deposited energy in the PMT is recorded.

Most of the hits are detected by the PMT, since even without the mirror, 98% of the interactions are recorded. Therefore, for a simple simulation set-up where no low threshold on the energy deposit is set, the mirror has a little effect on the results. However, the energies are shifted down to lower values without the mirror. For real measurements, this is very important, for a too low signal will not be detected (because of the electronic noise, the background etc.).

This feature, as well as the smearing out of the deposited energy, does not need to be simulated by Geant4. It is much less time consuming to simply modify the results given by simulations involving Compton scattering and photo-absorption.

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Chapter 4

Detector characterization for an

on-axis beam

In this chapter, realistic models of the two designs of concern in this thesis will be considered. The processes and particles involved in the simulations will be the same as in section 3.1, that is, only the mechanisms inherent to the detector function are taken in account. The materials used have all been tested before: these are vacuum, CsI and vinyl-toluene. Following the conclusion of section 3.2, the PMTs shall not be included in the Geant4 code.

4.1 GAP-like model

4.1.1 Modulation curve

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Figure 4.1: Numbering of the CsI scintillators of the GAP-like design.

An on axis 50 keV mono-energetic beam is directed at the detector. The beam cross section is circular and reaches the whole surface of the detector (plastic and CsI scintillators). Only the events consisting of one hit in the plastic scintillator and one hit in one CsI scintillator are taken into account. The resulting angular distribution is shown in Figure 4.2 for an unpolarized beam, and in Figure 4.3 for a 100% polarized beam.

As expected, the azimuthal angular distribution is constant for the unpolarized beam and exhibits a sinusoidal modulation for the polarized beam. In this later case, the modulation curve is fitted with a sinusoidal function. The modulation factor is read from the fitting parameters: µ100= 0.38±0.003.

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Figure 4.3: Azimuthal distribution for a 100% polarized beam, fitted with a sinusoidal function.

An other parameter of interest is the efficiency η. It is the ratio of valid two hits polarization events/number of particles hitting the detector. It is directly linked to the effective area Aef f:

Aef f = η × A, where A is the detector’s area. In our example, there are 163 893 events in total,

for 1 000 000 incoming photons, and the detector surface is (6 cm)2_{π = 130 cm}2_{. Hence:}

η = 163893

1000000 = 0.16; Aef f = 130 cm2× η = 21 cm2

4.1.2 Energy dependence of µ

100

and η

In order to assess the detector performance, it is important to know its energy dependence. The set-up used in the previous section is repeated for energies between 10 keV and 500 keV. The results are shown in Figure 4.4 and Figure 4.5.

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Figure 4.4: Efficiency for valid two-hits events as a function of the incoming X-ray energy calculated by Geant4 simulation.

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4.1.3 Influence of the low energy threshold

As explained in chapter 2, in order to prevent events from being triggered by electronic noise, it is necessary to set an energy threshold for each hit. The value of this threshold depends on the scintillator material, on the PMTs and on the quality of the electronics. It can be found only experimentally, but previous experiments (41) show that it lies between 1 keV and 10 keV for the plastic scintillators and around 1 keV for the CsI scintillators.

The results of the previous section simulations are reanalyzed with different lower energy thresh-olds for the plastic and CsI scintillators. The modulation factor and efficiency dependence on the energy are shown in Figures 4.6, 4.7, 4.8 and 4.9.

The low energy threshold in the plastic scintillator causes the efficiency to decrease and the maximum efficiency to be shifted towards higher energies. This was expected, since according to equation 1.2.4, the deposited energy is proportional to the incoming photon energy. Therefore low energy incoming photons are more influenced by a low energy threshold. When the low energy threshold is increased, the modulation factor maximum is also shifted towards higher energies, and it also increases, as shown in Figure 1.8.

The low energy threshold in the plastic scintillator does not modify the detector performance at high energy, but dramatically affects the performance at low energy. It can therefore decrease the detector energy range. The low energy threshold in the CsI scintillators on the other hand only has a very small impact on the results.

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Figure 4.7: Modulation factor as a function of the incoming X-ray energy for different low energy thresholds in the plastic scintillator, calculated by the Geant4 simulation. A hit is taken in account only if its deposited energy is higher than the threshold.

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Figure 4.9: Modulation factor as a function of the incoming X-ray energy for different low energy thresholds in the CsI scintillators, calculated by the Geant4 simulation. A hit is taken into account only if its deposited energy is higher than the threshold.

4.1.4 PMT effects

It has been showed in chapter 3 that the the main effect of the light collection and amplification by the PMT is to smear out the deposited energy according to a Poisson law and that it is much less time consuming to add this effect to the data during the analysis rather than to simulate it with Geant4.

The effects of the light transportation processes inside a scintillator and the generation and multiplication of photoelectrons in the photomultiplier tube on the deposit energy spectrum are well known. In e.g. (37), it is shown that in a plastic scintillator the light yield depends on the incoming photon energy, and that the deposited energy read out by the PMT is smeared out by a Poisson or a Gaussian law, depending on the energy. The light yield dependence on the energy and the broadening have to be tuned according to observations. Therefore, this paper’s results cannot be used for our models.

As mentioned in chapter 3, the information about the scintillator used as a model for our simu-lations only gives a constant light yield. The implementation of the PMT’s effects will be limited to a Poisson driven broadening of the energy spectrum. The deposited energy given by the Geant4 simulation is converted into the number of scintillation photons with the light yield. This number of photons is smeared with a Poisson distribution, and then reconverted into a deposited energy, which is used in the analysis.

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neither the modulation factor, nor the effective area, since it is only if and where a hit takes place that matters for those two parameters. But, if there is a lower energy threshold implemented in the plastic scintillator, some hits that would have been accepted will be rejected because the smearing out process has decreased their energy. Figures 4.10 to 4.15 show the efficiency and the modulation factor as a function of the incoming X-ray energy with and without smearing out of the deposit energy. At low energy, it causes the efficiency to decrease, and the modulation factor to increase, both very slightly. At high energies, there is essentially no change at all.

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Figure 4.11: Modulation factor as a function of the incoming X-ray energy, calculated by the Geant4 simulation.

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4.1.5 Comparison with GAP results

Figure 4.16: The GAP geometry as presented in (9).

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for the GAP detector. The low energy thresholds for the CsI and the plastic scintillator used for these simulations are given in (9): 8 keV for the plastic scintillator and 30 keV for the CsI scintillators. It is also said that a upper energy threshold is set, but there is very few details about it: ”≈ 220 keV”. In Figure 4.18 and Figure 4.19 the GAP results are compared with our simulations results, for different high energy thresholds.

The energy dependence of the efficiency of our simulations agrees with the GAP one for energies below ≈80 keV. For energies between 80 keV and 220 keV, GAP efficiency is slightly greater, independently of the settings of our simulations. However, if these latter are well tuned, our simulations’ curve converges again with the GAP’s one for energies greater than 220 keV. The geometry of GAP is never precisely defined. In particular, the proportion of the plastic central scintillator which is conic is not given. This could be a source of the divergence between our results and the GAP’s one. It is not said if CsI-CsI events are taken into account in GAP’s modulation curve. For the photo-absorption in the CsI happens at higher energy than in the plastic scintillator, this could explain the divergence at intermediate energies.

The energy dependence of the modulation is greater for GAP than for our simulations, for all energies. For energies above ≈70 keV, and for certain settings, the shape of the curves are in agreement. For energy lower than 70 keV however, the behaviors of the two models are dramatically different. Although the simulations presented in (9) are said to be run with low energy thresholds of 8 keV for the plastic scintillator and 30 keV for the CsI scintillators, they match better at low energy with our simulations when no thresholds is set. Moreover, a earlier version of (9), whose results are shown in Figure 4.20, exhibits a modulation factor with a behavior closer to ours. The GAP modulation factor is clearly greater than in our simulations. This divergence might arise form different selection of the events. In our model, the CsI-CsI events have not been taken into account in the calculation of the modulation factor nor of the efficiency. This type of event has a better angular resolution than the Plastic-CsI type and would therefore increase the total modulation factor.

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Figure 4.18: Efficiency as a function of the incoming X-ray energy. The low energy thresholds for the GAP’s simulations are 8 keV for the plastic scintillator and 30 keV for the CsIs. The high energy thresholds are not known. Our simulations have the same low energy thresholds as GAP, and different high energy thresholds are simulated. They are given in parentheses.

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Figure 4.20: Energy dependence of the modulation factor and the efficiency of the GAP detector according to the simulations presented in an earlier version of (9).

4.2 GRAPE-like model

4.2.1 Modulation curve

For the GAP-like model, the modulation curve could be obtained by plotting the number of inter-actions with respect to the detector’s number (as shown in Figure 4.1). For the GRAPE-like model it is not as straightforward, and the angle between two scintillators must be explicitly computed. The technical details were given in chapter 2.

A mono-energetic, unpolarized, on axis beam is directed at the detector. The beam cross section is a square so that all photons reach the detector and the entire face of the detector is illuminated (plastic and CsI scintillators). Only events consisting of one hit in a plastic scintillator and one hit in one CsI are recorded. Figure 4.21 shows the resulting angular distribution. Unlike for the GAP-like model, it is not constant. This is due to the non circular symmetry of the detector itself. For example, in each scintillator, the horizontal and vertical directions are privileged over the diagonal ones, since to reach another scintillator in those directions, and therefore be detected, a photon needs to travel further. The counting rate exhibits a square symmetry, which stems from the shape of the detector.

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possible solution is to rotate the detector while it acquires data. Unfortunately, this is not feasible for GRBs, since the observation time is too short. Another possibility, described by (3) consists in renormalizing the angular distribution by the unpolarized angular distribution, as shown in equation 4.2.1.

Nnorm(Φ) =

Npol(Φ)

Nnon(Φ)

× Nmax (4.2.1)

where Nnormis the renormalized distribution, Npolthe distribution for a polarized beam, Nnonthe

distribution for an unpolarized beam, and Nmax the maximum of Nnon.

The results of this method are shown in Figure 4.23. The fit after renormalization indicates a modulation factor µ = 0.61, with a relative error of 1%.

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Figure 4.22: Azimuthal angular distribution for a 100% polarized beam. Geant4 simulation with one million 100 keV incoming photons.

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4.2.2 Energy dependence of µ

100

and η

As for the GAP-like design, the modulation factor and the efficiency are systematically studied for a large set of energies. For each simulation, a beam of 10 million photons is directed at the detector. The resulting curves are shown in Figure 4.24 and Figure 4.25.

The efficiency reaches a maximum for E≈50 keV and it is optimized around 30-150 keV. It decreases slowly at high energies and drops just below 0.06 at 500 keV. The modulation factor goes as high as 0.7 for an energy around 50 keV, and it drops then slowly to reach 0.5 for 500 keV. These results show that the detector’s energy range is limited by its efficiency and not by its modulation factor. This is an interesting feature, for the modulation factor is inherent to the detector’s design, whereas the efficiency can be increased for instance by increasing the number of modules.

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Figure 4.25: Efficiency dependence on the energy for 100% polarized beams.

4.2.3 Influence of the low energy threshold

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Figure 4.26: Efficiency for different low energy threshold in the plastic scintillators. Each simulation involves one million incoming photons.

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Figure 4.28: Efficiency for different low energy threshold in the plastic scintillators. Each simulation involves one million incoming photons.

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4.2.4 PMT effects

As for the GAP-like model, the implementation of the PMT effects is limited to a Poisson law broadening of the energy. The data from the previous simulations are reanalyzed after the en-ergy of all the hits has been randomly smeared out. The results are shown in figures 4.30, 4.31, 4.32 and 4.33. As for the GAP-model, the energy broadening has practically no influence on the modulation factor nor the efficiency.

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Figure 4.31: Modulation factor dependence on the energy. Blue curve: no threshold nor smearing applied. Red curve: the plastic lower energy threshold is 1 keV, but no smearing of the data. Green curve: the plastic lower energy threshold is 1 keV and the data are smeared.

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Figure 4.33: Modulation factor dependence on the energy. Blue curve: no threshold nor smearing applied. Red curve: the plastic lower energy threshold is 10 keV, but no smearing of the data. Green curve: the plastic lower energy threshold is 10 keV and the data are smeared.

4.2.5 Discrimination of different types of events

As pointed out in chapter 2, different types of events can be taken in account when the angular distribution is calculated. Figure 4.34 shows the efficiency of each type of events.

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Figure 4.34: Efficiency dependence on the energy for different types of event.

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Figure 4.35: Modulation factor dependence on the energy for 100% polarized beams.

CsI-CsI: The CC events have a higher modulation factor than the PC events. Thus, taking them in account increases both the efficiency and the modulation factor of the detector. In Figure 4.35, the modulation factor for CC events is not computed at low energy. That does not mean that the CC events should not be taken in account at low energy, since in real measurement, the modulation factor will be computed using all the events types together.

Plastic-Plastic: At higher energy than 150 keV, the PP modulation factor is lower than the CP modulation factor. Moreover, the efficiency of PP events is rather low (e.g in comparison with the CC one). Therefore it is no obvious whether or not the PP events should be taken in account.

4.2.6 Cross-talk and neighbouring hits

In the GRAPE detector, a multi-anode PMT (MAPMT) is used to read the signals form the plastic scintillators. Although it is not simulated in our Geant4 simulations, in reality it can be affected by crosstalk. That is, a single hit is detected by two neighbouring anodes of the MAPMT and thus it seems that there is one hit in each of the two neighbouring scintillators read out by these anodes. Hence, a double hit event is registered, whereas only a single hit event happened.

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Figure 4.36: Efficiency for the Plastic-Plastic events.

Figure 4.37: Modulation factor for the Plastic-Plastic events.

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factor increases (the excluded event have the lowest angular resolution). The change is very small, especially for the modulation factor. Figures 4.38 and 4.39 show how the exclusion of neighbouring hits (for PP events) influences the total efficiency and modulation factor. The modulation factor is practically not affected. The efficiency on the other hand decreases by ≈ 5%.

Rejecting the neighbor hits diminished the performance of the detector. But this conclusion is not relevant since the point of this exclusion is to compensate the effects off crosstalk, which is not simulated. It would be interesting to further investigate this phenomenon, with simulations, experiments or theoretical models. But this is beyond the scope of this thesis.

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Figure 4.39: Modulation factor for PP, CC and PC events.

4.2.7 Influence of the polarization angle

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Figure 4.40: PC-events modulation factor dependence on the energy for different polarization angles (θ).

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Figure 4.42: CC-events modulation factor dependence on the energy for different polarization angles (θ).

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For PC and CC events, the modulation factor is not significantly influenced by the polarization angle. For the PP events on the contrary, the difference between several polarization angles goes up to 10%.

4.2.8 Comparison with GRAPE results

It is interesting to compare our results to the GRAPE ones (36). Figure 4.44 displays the GRAPE effective area. In order to compare to our GRAPE-like model efficiency, the effective area must be divided by the GRAPE area (16 cm2_{). The expected performance for the GRAPE detector has}

been presented in (36). In particular, the effective area and modulation factor dependence on the energy have been simulated with Geant4. The results are shown in Figures 4.44 and 4.45.

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Figure 4.45: Simulation results for a single module of the GRAPE detector showing the modulation factor as a function of energy for the various event types (36).

The ratio between the different types of event efficiency is similar for the two models. Figure 4.46 compares the GRAPE and our model efficiencies for different settings. At high energy, our model has a higher overall efficiency, regardless of the setting chosen. At low energy, the two curves match for certain low energy thresholds. Figure 4.47 compares the modulation factor dependencies to the energy. For the total and the PC modulation factors, the two models are in good agreement. The CC events modulation factor is slightly higher for our model, than for GRAPE. The results for the PP modulation factor shows a clear divergence, regardless of the settings chosen.

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(a) PC events (b) PP events

(c) CC events

Figure 4.47: Comparison of the modulation factor dependence on the energy for GRAPE (36) and our GRAPE-like model. The different types of events are shown separately.

4.3 Conclusion

Figure 4.49 compares the like and the GAP-like modulation factors. For the GRAPE-like model, it is more than twice as high as for the GAP-GRAPE-like model. This is expected since the GRAPE-like detector has a better angular resolution.

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scattering mean free path lmeanis given in equation 4.3.1.

lmean(E) =

1 ne× σ(E)

, (4.3.1)

where ne is the electronic density and σ the Compton cross section lmean is higher for low energy

photons. Therefore, the efficiency will be more affected by geometrical limitations at low energy. As shown in Figure 4.48, photons path is more limited by the GAP-like design than by the GRAPE-like one. Thus, the divergence between the two models’ efficiency should be greater at low energy, which is what is observed.

Figure 4.48: ”Available” photon path in GRAPE and GAP like designs. on average, it is less for Gap.

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Figure 4.50: Comparison of the GRAPE’s and GAP’s efficiency. The GRAPE’s efficiency is shown with and without taking into account the PP and CC-events.

Although the comparisons of both modulation factors and efficiencies seem to indicate that the GRAPE-like model is better than the GAP-like model, this conclusion must be put into perspective. First of all, when comparing the two models’ efficiencies, the weight of the models should be accounted for, for it is a limitation in satellite borne mission. For instance the conical shape of GAP plastic scintillator lowers its efficiency, but it also lowers its weight. This can be taken into account by rescaling the efficiency with the ”vertical density”, that is the surface of the detector area divided by its weight. It can be approximated by the ratio area/volume.

For the GRAPE-like detector:

Area Volume =

25 cm2

125 cm3 = 0.2 cm

−1 _(4.3.2)

For the GAP-like detector:

Area Volume =

130 cm2

430 cm3 = 0.3 cm

−1 _(4.3.3)

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Figure 4.51: Comparison of the GRAPE’s and GAP’s efficiency renormalized by the detector’s weight. The efficiencies are computed for PC-events only.

Secondly, the study of CC-events should be done for the GAP-like detector too, for it might increase its modulation factor and it would allow the overall efficiency of the two detectors to be compared. Moreover, it must be taken in account that the GAP-like model circular symmetry is a strong advantage. The GRAPE-like model geometry causes fake modulations, which makes the analysis more complicated. The statistical implication of these should be thoroughly investigated. Such an investigation is, however, beyond the scope of this thesis.

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Chapter 5

Detector characterization for a

realistic GRB spectrum

In this chapter, the GAP-like model will be tested under more realistic conditions. That is, its performance when a off axis beam with a GRB (gamma-ray burst) spectrum is directed at it will be assessed. Our choice of GRB spectrum will be first described; it will be then discussed how to treat off axis events and finally, the results obtained will be presented. This chapter also provides an outlook towards the next step of this project.

5.1 GRB spectrum

5.1.1 A realistic GRB spectrum

A widely used function for fitting GRB spectra is the Band function (38), given in equation 5.1.1:

fBAN D(E) = A    E 100 keV α exph−(α+2)E_E peak i , E <(α−β)Epeak α+2 E 100 keV β

exp(β − α)h (α−β)Epeak

100 keV(α+2)

iα−β

, E ≥(α−β)Epeak

α+2

(5.1.1)

The four free parameters are the amplitude, A, the low and high energy spectral indices, α and β respectively, and the peak energy, Epeak. This function can be approximated by a broken power

Simulation studies for an X-ray polarization satellite Master of Science Thesis