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Applications of Pulse Shape Analysis Techniques for

Segmented Planar Germanium Detectors

ANTON KHAPLANOV

Licentiate Thesis in Physics

Stockholm, Sweden 2007

TRITA FYS 2007-80ISSN 0280-316XISRN KTH/FYS/–07:80–SEISBN 978-91-7178-837-5

KTH PhysicsSE-106 91 Stockholm

SWEDEN

Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framläggestill offentlig granskning för avläggande av teknologie licentiatexamen i fysik torsda-gen den 13 december 2007 klockan 14.00 i sal FB53, AlbaNova Universitetscentrum,Stockholm.

© Anton Khaplanov, December 2007

Tryck: Universitetsservice US-AB

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Abstract

The application of pulse shape analysis (PSA) and γ-ray tracking techniqueshas attracted a great deal of interest in the recent years in fields ranging fromnuclear structure studies to medical imaging. These new data analysis meth-ods add position sensitivity as well as directional information for the detectedγ-rays to the excellent energy resolution of germanium detectors. This thesisfocuses on the application of PSA on planar segmented germanium detectors,divided into three separate studies. The pulse shape analysis technique knownas the matrix method was chosen due to its ability to treat events with ar-bitrary number and combinations of interactions within a single detector. Ithas been applied in two experiments with the 25-fold segmented planar pixeldetector – imaging and polarization measurements – as well as in a simulationof upcoming detectors for DESPEC at NuSTAR/FAIR.

In the first experiment, a point source of 137Cs was imaged. Events wherethe 662 keV γ-rays scattered once and were then absorbed in a different seg-ment were treated by the PSA algorithm in order to find the locations of theseinteractions. The Compton scattering formula was then used to determine thedirection to the source. The experiment has provided a robust test of the per-formance of the PSA algorithm on multiple interaction events, in particularthose with interactions in adjacent segments, as well as allowed to estimatethe realistically attainable position resolution. In the second experiment, theresponse of the detector to polarized photons of 288 keV was studied. Thepolarization of photons can be measured through the observation of the an-gular distribution of Compton-scattered photons, Hence the ability to resolvethe interaction locations had once again proven useful.

The third study is focused on the performance of the proposed planargermanium detectors for the DESPEC array. As these detectors have notyet been manufactured at the time of this writing, a set of data simulatedin GEANT4 was used. The detector response was calculated for two of thepossible segmentation patterns – that with a single pixelated contact andone where both contacts are segmented into mutually orthogonal strips. Inboth cases, PSA was applied in order to reconstruct the interaction locationsfrom this response. It was found that the double-sided strip detector canachieve an over-all better position resolution with a given number of readoutchannels. However, this comes at the expense of a small number of complexevents where the reconstruction fails. These results have also been comparedto the performance of the 25-fold pixelated detector.

List of Papers

This thesis is based on the following papers.

1. A. Khaplanov, J. Pettersson, B. Cederwall, Compton imager based on a single

planar segmented HPGe detector, Nucl. Instr. and Meth. A580 (2007), 1075.

2. A. Khaplanov, S. Tashenov, B. Cederwall, G. Jaworski, A gamma-ray po-

larimeter based on a single segmented planar HPGe detector, submitted toNucl. Instr. and Meth. A.

3. A. Khaplanov, B. Cederwall, S. Tashenov, Position sensitivity of segmented

planar HPGe detectors for the DESPEC project at FAIR, submitted to Nucl.Instr. and Meth. A.

The author performed most of the work contributing to the above papers. Con-tributions by others include the following. The image reconstruction as well as theMonte Carlo simulation for the experiment in paper 1 were done by J. Pettersson.The analysis of data for paper 2 was performed in collaboration with S.Tashenov.In the third paper, the Monte Carlo simulation of the interactions of photons withthe detector were performed by S. Tashenov.

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Contents

List of Papers v

Contents vi

1 Introduction 1

1.1 Photon interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Photoelectric absorption . . . . . . . . . . . . . . . . . . . . . 21.1.2 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.3 Pair production . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Semiconductor detectors . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 KTH planar segmented detector . . . . . . . . . . . . . . . . 4

1.3 Pulse shape analysis and γ-ray tracking . . . . . . . . . . . . . . . . 51.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.1 New generation of γ-ray spectrometers . . . . . . . . . . . . . 51.4.2 γ-ray imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Pulse Shape Analysis 9

2.1 Physics of signal formation in germanium detectors . . . . . . . . . . 92.1.1 Position sensitivity and segmented detectors . . . . . . . . . . 10

2.2 Simulation of pulse shapes . . . . . . . . . . . . . . . . . . . . . . . . 122.2.1 Electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 The weighting field method . . . . . . . . . . . . . . . . . . . 122.2.3 Detector characterization . . . . . . . . . . . . . . . . . . . . 13

2.3 Pulse shape analysis methods . . . . . . . . . . . . . . . . . . . . . . 132.3.1 Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 The matrix method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5 The matrix method: in-depth . . . . . . . . . . . . . . . . . . . . . . 17

2.5.1 Solving the matrix equation . . . . . . . . . . . . . . . . . . . 172.5.2 The definition of the interaction points . . . . . . . . . . . . . 20

3 Experimental applications 23

3.1 Compton Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1.1 Imaging with the planar HPGe detector . . . . . . . . . . . . 24

vi

Contents vii

3.1.2 Comparison with simulation . . . . . . . . . . . . . . . . . . . 253.1.3 Imaging resolution and efficiency . . . . . . . . . . . . . . . . 27

3.2 γ-ray polarimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.1 Polarization measurement . . . . . . . . . . . . . . . . . . . . 293.2.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2.3 PSA versus pixels . . . . . . . . . . . . . . . . . . . . . . . . 31

4 DESPEC planar detectors 35

4.1 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.1.1 Segments and voxels . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Choice of the basis grid . . . . . . . . . . . . . . . . . . . . . . . . . 394.3 PSA results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.4 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

A Digital data acquisition 47

B Glossary 49

Acknowledgements 51

Bibliography 53

Chapter 1

Introduction

Germanium detectors are the main tools in today’s γ-ray spectroscopy. Combin-ing excellent resolution with large sensitive volume and a high stopping power,the high purity germanium (HPGe) detectors are found in most nuclear physicsexperiments. It has been found that a two-dimensional segmentation of the detec-tor surface enables a three-dimensional position sensitivity through the analysis ofpulse shapes [1, 2]. This feature is highly desirable in nuclear physics experimentsas it has been found that it can be used to greatly increase the efficiency of a spec-trometer [3]. It also opens the possibilities of new applications where previouslyonly highly granulated detectors or detector systems with large amount of collima-tor material were used. In particular, γ-ray imaging applications, including thosewith polarization sensitivity, are of great interest in today’s medicine, astronomyand nuclear non-proliferation safeguards.

1.1 Photon interactions

A photon traversing matter has several interaction possibilities. Unlike chargedparticles, a photon suffers no continuous energy loss but looses energy at discreteinteraction points. The cross sections of the possible interactions in germaniumas a function of the γ-ray energy are presented in fig. 1.1. We see that Compton(or incoherent) scattering and photoelectric absorption are the two dominatingprocesses for typical γ-ray energies of 100 keV–2 MeV. The majority of interactionsconsist of a series of scatterings followed by a final photo-absorption.

The fact that all of the interaction points of a γ ray may or may not be insidethe sensitive volume of a detector presents difficulties in measuring its total energy– for incomplete energy collection, an event contributes to the so-called Comptondistribution in a spectrum. At the same time, the kinematics of the interactionsare not random, and with advanced analysis, this difficulty can be turned into anadvantage.

1

2 CHAPTER 1. INTRODUCTION

Figure 1.1: Photon cross section in Germanium [4]

1.1.1 Photoelectric absorption

A photon may transfer all of its energy to an atomic electron. This is knownas photo absorption. Fig. 1.1 shows that the cross section for this process growsrapidly as the energy decreases. Since the electrons are initially bound, a part ofthe photon energy is spent to free the electron and the rest is transferred to theelectron as kinetic energy. If Ebind is the binding energy and hω is the energy ofthe photon, the kinetic energy of the electron is given by

Ekin = hω − Ebind, (1.1)

The characteristic edges in the photoelectric cross section are found at the bindingenergies of the atomic electrons, where the photon energy becomes insufficient tofree the electrons of a particular shell.

A photoelectric interaction manifests itself as a point-like interaction in thecontext of large-volume solid-state detectors. Both components of the depositedenergy - the fast electron and the x-ray photon (or an Auger electron) created

1.1. PHOTON INTERACTIONS 3

when the vacancy left by the electron is filled - are typically stopped within 1 mmor less from the interaction point.

1.1.2 Scattering

Compton, or incoherent, scattering is the dominant mode of interaction for photonsin germanium for energies between 150 keV and 8 MeV, see fig. 1.1. In a Comptonscattering, the energy of an incoming photon is partitioned between a scatteredphoton and an atomic or a free electron. Due to the conservation of momentum,the momentum vectors of the initial photon, the final photon and the recoilingelectron lie in a plane. The initial energy of the photon, Eγ , is shared between thescattered photon, E′γ , and the electron, Ee− according to

E′γ =Eγ

1 +Eγmec2

(1− cos θ)(1.2)

Ee− =

E2

γ

mec2(1− cos θ)

1 +Eγmec2

(1− cos θ)(1.3)

The energy transferred to the electron increases with the scattering angle, θ, andis greatest in the case of backscattering (θ = 180). It is also generally true thatthe typical energy transfer increases with the photon energy. It is common thata photon undergoes Compton scattering several times until its energy is reducedsufficiently for photo-absorption to become the more likely interaction.

In the classical limit, Eγ << mec2, scattering on free electrons, this reduces

to Thomson scattering. The scattering of a photon against an atom as a whole isknown as Rayleigh or coherent scattering and is also primarily important for lowenergies.

1.1.3 Pair production

In a pair production, a photon interacts with the electric field of a nucleus oran electron creating an electron and positron pair. As can be seen in fig. 1.1,the interaction with a nuclear field is far more likely. Unlike the other types ofinteractions, pair production has an energy threshold at twice the electron restmass required by energy conservation. Any energy over 1022 keV is shared betweenthe electron and the positron. Both particles are stopped in the vicinity of theinteraction point. The stopped positron annihilates with an electron in the material,usually producing a pair of 511 keV photons. These generally need not interact closeto the pair production point but will undergo a series of Compton scatterings andeventually a photo absorption in the same way as any other photons, and may alsoescape from the detector volume.


1.2 Semiconductor detectors

Many types of semiconductor detectors exist. Silicon is commonly used for the de-tection of charged particles and low-energy photons. The detectors may be arrayedas thin wafers intended to track a particle as it traverses one wafer after the other.In applications where a particle’s total energy is of interest, it is often implanted ina thicker silicon detector and stopped in it. Germanium is the material of choicefor the detection of γ rays due to its greater atomic number and density. Bothlarge-volume silicon and germanium photon detectors require cryogenic cooling inorder to suppress the thermal noise. The thin silicon wafer used for charged particledetection may be operated at room temperature.

1.2.1 KTH planar segmented detector

A planar pixelated detector at KTH was used in the experimental parts of thiswork. The detector is a 58×58×21 mm high purity germanium (HPGe) crystal.The cathode contact has a 4-mm guard ring and a 5×5 pixel segmentation, witheach pixel having the area of 1 cm2, see fig 1.2. The anode contact is not segmentedand covers the entire face of the crystal. The signals are read out by charge-sensitive preamplifiers and digitized using the data acquisition system described inappendix A.

−20−10

010

20

−20

−10

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10

20

XY

Z

Figure 1.2: The geometry of the 25-pixel KTH planar detector.

1.3. PULSE SHAPE ANALYSIS AND γ-RAY TRACKING 5

1.3 Pulse shape analysis and γ-ray tracking

The goal of pulse shape analysis (PSA) and γ-ray tracking is the reconstruction ofpaths taken by photons interacting with a detector on an event-by-event basis. Theseries of interactions of the types described above constitute a track of point-likeenergy deposits. For the γ-ray energies typical for nuclear physics (∼100-2000 keV),the tracks will predominantly consist of several Compton scattering points anda final photo-absorption. Pair production becomes significant towards the high-energy end of the spectrum. In the events where pair production does take place, itis almost certainly the first interaction and the remaining ones constitute the tracksof the two annihilation photons. Coherent scattering is normally not essential forthese energies.

The task of determining the track of a γ ray is divided into two distinct problems.The first is to find the points within the detector where energy has been deposited.This is accomplished by the PSA algorithms. The second problem is to determinethe order in which the points found by PSA were visited, and in more complexcases with many interacting photons to find which interactions belong to the trackof a given photon and which do not. The main tool used by the tracking algorithmsis the Compton scattering formula 1.3 due to its ability to relate the geometricallocations of interactions to the deposited energy. In many applications it proves tobe useful to have information concerning the direction of the incoming photon oreven just the position of its first interaction with the detector.

1.4 Applications

1.4.1 New generation of γ-ray spectrometers

Two new γ-ray spectrometers for nuclear structure studies, AGATA [5] in Europeand GRETA [6] in the USA, are currently being constructed. Each spectrometeremploys PSA and tracking to overcome many limitations of spectrometers of theprevious generations.

The currently operating arrays such as JUROGAM [7] and EXOGAM [8] rely ongermanium detectors to obtain the γ-ray spectra. Each detector is surrounded by aset of scintillator detectors functioning as an anticoincidence shield that rejects theevents where photons escape from germanium before a complete absorption. Thistechnique greatly reduces the Compton background in the spectra. Unfortunatelyit also rejects a large portion of the γ-rays. Furthermore, the Compton shieldsoccupy much of the solid angle seen by the γ-rays, thus further limiting efficiency.

The new arrays seek to overcome these limitations by using very closely-packedgermanium detectors with tracking capability, eliminating the need for escape sup-pression. The schematic of the geometry of AGATA is shown in fig. 1.3 – the 180detectors needed to complete this array form a near-4π coverage. The photons thatscatter in one detector and escape need no longer be rejected – they are likely to beabsorbed in a neighboring one. Pulse shape analysis finds individual interactions,


and γ-ray tracking is not restricted to a single detector. The tracking algorithmsare also intended to disentangle a large number of interactions that may occursimultaneously as a result of a prompt γ-ray flash.

Figure 1.3: Schematic of the closely packed germanium crystals for AGATA.

A further limitation of the conventional arrays is the Doppler broadening ofthe spectral lines that occurs in the experiments where the emitting nucleus has alarge velocity during the decay. The Doppler-shifted energies can be corrected ifthe emission direction with the respect to the velocity vector is known, however,the large volume of the germanium detectors limits the attainable precision. Theadvantage of a tracking array lies in its ability to locate the first interaction point,allowing to perform a finer Doppler correction. This capability has been testedusing the AGATA prototype detectors during an in-beam test [9].

The DESPEC array is proposed for the spectroscopy of the decaying recoilsimplanted at the focal plane of the Super-FRS at the upcoming FAIR facility [10].This detector array will take the advantage of tracking in order to increase efficiencysimilarly to AGATA, as well as to find the direction to the source of the radiation,enabling the rejection of background that is not originating at the implantationdetector.

1.4.2 γ-ray imaging

As has already been mentioned, a tracking algorithm is capable of estimating thedirection to the source. In practice, this direction will have a rotational degree offreedom, forming a cone of possible directions to the source. Nevertheless, suchcones from many interactions can combine to an image. This is known as Compton

1.4. APPLICATIONS 7

imaging and has been applied primarily in γ-ray astronomy [11]. Conventionally,one relies on highly granulated detector systems, segmented germanium detectors,however, may allow imaging using only two or even a single detector [12, 13, 14].

In addition to γ-ray astronomy, γ-ray imaging is a highly important tool for non-invasive medical examinations. Both the gamma camera, as well as the positronemission tomograph (PET) may benefit from the advent of position-sensitive de-tectors [15]. In the recent years there has also emerged a significant interest forγ-ray imaging employed for national security and nuclear non-proliferation safe-guards [12, 16].

Chapter 2

Pulse Shape Analysis

The term pulse shape analysis in the context of germanium detectors refers to thelocation of the individual interaction points of a γ ray. In segmented large-volumeHPGe detectors PSA has enabled a resolution of the positions of the interactionpoints that is about one order of magnitude more accurate than can be given bythe dimensions of the physical segmentation [2, 17]. This chapter summarizes thephysics responsible for the position sensitivity in segmented detectors as well aspresents the methods of pulse shape analysis. In particular, the matrix method [23]employed in this work is presented in detail.

2.1 Physics of signal formation in germanium detectors

All types of interaction of a γ ray with the detector material – be it Comptonscattering, photo absorption or pair production – liberate charge in the form ofenergetic electrons (or, in the case of pair production also positrons). These quicklyloose their energy to the surrounding material through ionization, bremsstrahlungand multiple scattering, thus ionizing the material. The charge cloud created insuch processes has dimensions of the order of 1 mm in Ge, and can in most casesbe treated as a point charge, provided that the detecting elements are much larger.

A high voltage, HV, typically ranging between 1 and 4 kV depending on thedetector geometry, is applied to two opposing electrical contacts of the detectorcreating an electric field throughout the detector. The magnitude of the HV isadjusted so that the semiconductor crystal is fully depleted, ensuring that the onlysource of free charges is interactions of radiation. Thermal excitation of chargesis largely suppressed by cooling the detector to the temperature of liquid nitrogen(77 K).

The applied electric field causes the charges to drift towards the contacts (bothholes and electrons travel to the contact of the opposite polarity). The dependenceof drift velocities on the applied field has been studied by eg. Mihailescu, et.al. [18].Germanium forms a face centered cubic (FCC) crystal lattice. The drift velocities

9

10 CHAPTER 2. PULSE SHAPE ANALYSIS

are not identical for the three crystallographic axes and may differ by as much asa factor of 1.3. Planar germanium detectors are normally cut so that the <100>axis is parallel to the applied field, in coaxial detectors, however, the motion of thecharges can occur in any direction with respect to the crystallographic axes.

The presence of charges causes induced charges to appear on the electrical con-tacts. As the charges drift, the magnitude of the induced charges changes andfinally reaches a maximum when the charges themselves reach the contacts. Theinduced charges are collected on a capacitor in a charge-sensitive preamplifier andit is essentially this signal that is read out in a measurement.

2.1.1 Position sensitivity and segmented detectors

As both the motion of electrons and of holes contribute to the formation of signals,the shapes of the pulses consist of two components. Consider a contact at a positivepotential. Electrons are attracted to this contact while the holes are repelled. Atthe moment of the creation of the charge cloud, both charges are in the samelocation and no charge is induced. As the electrons drift towards the contact theircontribution to the signal grows while the contribution of the holes diminishes,thus the charge drifting towards a contact is said to dominate the signal. A similarsituation is seen on the opposite contact, with only the polarity reversed.

This symmetry is broken if the interaction point lies closer to one contacts thanthe other. Say the interaction is close to the positive contact. The electrons arethen quickly collected contributing to the greater part of the signal. The holes onthe other hand recede and contribute less and less as time passes. In other wordsthis creates a signal whose leading edge rises quickly in the beginning and slowlythereafter. The situation on the negative contact is different - the quick collectionof the electrons on the far-away contact is largely unnoticed, while the holes’ con-tribution increases gradually until maximum. Clearly the information about theshapes of the pulses may be used to determine the depth of the interaction, in facta very simple timing algorithm can produce good results [14, 17]. The positions inthe other two coordinates can be obtained by electrically segmenting the contacts.

While it is possible to segment the contacts in two dimensions producing asufficient granularity to match the position resolution in the depth coordinate, itis not necessary – a rather small number of contacts in combination with PSA canprovide an equally good position resolution. The induced charge is distributed overthe contacts - the higher charge density is of course found in the area closest tothe interaction point, yet the entire surface will sense some signal. If the contactis broken up into electrically insulated segments, the induced charge is similarlydivided. Unless the charge cloud is exactly at the border of two segments, onlyone collects the net charge. The neighboring contacts sense a smaller amount ofinduced charge and this charge will be present only during the drift time. Oncecollected at one of the segments the charge can no longer induce signals in the othersegments. This type of signals is known as mirror or transient signals.

Figure 2.1 shows two examples of possible pulse shapes. In general, the closer

2.1. PHYSICS OF SIGNAL FORMATION IN GERMANIUM DETECTORS 11

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Figure 2.1: Pulse shapes from two single interactions in the 25-pixel detector. Theinteractions are at the same x and y coordinate but at different depth – 2 mm (blue)and 18 mm (green) from the segmented contact. The pulses from one interactionare shown in blue and from the other in green at the positions of the respectivepixels. The segment Q12 in this case collects the net charge – note that it is theonly one whose signal does not return to zero after the charge collection is complete.


a segment is to the interaction point the more prominent is its transient signal.Furthermore, rise time, time to maximum and the polarity of the transient signalschanges with the depth of the interaction. A similar, although a less intuitive effectcan be seen in a coaxial geometry where the outer contact is segmented. All of thisinformation combined can be used to obtain the three-dimensional position of theinteraction points of a γ ray. Typically, PSA can be expected to produce a positionresolution that is an order of magnitude finer than the physical segmentation [1, 2].

2.2 Simulation of pulse shapes

The pulse shapes arising from interactions of photons at arbitrary positions in thedetector volume can be calculated. While for a few simple geometries it is possibleto calculate an approximation analytically [1], in general, numerical methods mustbe used. A dedicated software was written for this task. The following outlines theprocedure used to calculate pulse shapes.

2.2.1 Electric field

First, the electric field throughout the detector must be calculated. The finiteelement method package, FEMLAB [19], was used, producing a database of val-ues of the electric field along an irregular grid of points throughout the detector.The density of the grid is increased in the areas near the segment edges where thefield gradients are greatest. The impurity concentration in the germanium crystalis taken into account by the calculation. Once the field is known, the trajecto-ries of holes and electrons are calculated in 1 ns steps, using the respective driftvelocities [18], until they reach the contacts.

2.2.2 The weighting field method

The problem of calculating a charge induced on a contact of arbitrary shape by apoint charge is a complex one. The solution to a relatively simpler problem, how-ever, provides the same result according to Ramo’s theorem [20]. The electrostaticcoupling between the moving charge and the sensing electrode is described by theweighting potential, and the corresponding weighting field. This field is calculatedfor each contact using the geometry of the detector as it is used to determine thereal electric field stemming from the applied high voltage, however, the potentialat the contact in question is set to unity and at all remaining contacts to zero.The space-charge distribution is also set to zero. A moving charge in the detectorvolume induces a current flowing in or out of each contact given by

I = qv · Ew, (2.1)

where v is the velocity of the charge and Ew is the weighting field at its currentlocation. In a way, the weighting field can be seen as a map of the sensitivity of agiven segment.

2.3. PULSE SHAPE ANALYSIS METHODS 13

X

Z

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9A B

Figure 2.2: The weighting potential for segment A calculated using FEMLAB.

An example of a weighting potential associated with a weighting field for oneof the contacts (segment A) of a planar detector is shown in fig. 2.2. A chargemoving along the z-axis towards the segment A experiences a continuously growingweighting field all the way to the contact - this is a charge signal. A charge movingtowards the neighboring segment B encounters a field that first increases and thenreduces to zero (the contact itself is set to ground) - this is a transient charge.Finally, the induced charge signals are obtained by folding the current signals withthe preamplifier response function.

2.2.3 Detector characterization

Most PSA algorithms rely on a set of accurate pulse shapes to be available for someinteraction locations (basis signals). These can be calculated as described above ormeasured using a detector scanning setup [1, 21]. Scanning a detector is a lengthyprocedure and for multi-detector systems such as AGATA it is unrealistic to scanall detectors, thus calculated signals can be used once they have been verified usingscanning data.

2.3 Pulse shape analysis methods

A variety of PSA methods are currently under development. A great source ofmotivation in this area are the AGATA and GRETA projects where PSA will bean integral part of the on-line data analysis. While it is beyond the scope of thisthesis to detail all of these methods, it is interesting to compare some of the ideasthat are being considered.

It is possible to generate pulse shapes for the interactions on a grid, chosen sothat the resulting pulses form a complete (or rather a sufficient) basis. Interactionsat arbitrary locations are then represented using either the closest basis point or


an interpolation between a set of basis points. The challenge is to find the corre-spondence between a measured pulse and a typically very large set of pre-computedbasis pulses. Grid search and particle swarm optimization are two examples of al-gorithms where a sub-set of the basis pulses is compared to the measured signalusing a χ2 criterion, resulting in a better-fitting sub-set until the search convergesto a single (or several) point(s) [9]. The matrix method, more thoroughly detailedbelow, considers linear combinations of possible basis points. It is also possible toperform a wavelet transformation of both basis and experimental signals and thenfit the wavelet coefficients [9]. An artificial intelligence technique, called genetic al-gorithm, has also been applied to pulse shape analysis [22]. This method treats theinteraction points as an evolving population, where individuals’ survival dependson their fitness, i.e. similarity to the measured signal.

Another approach relies on parametrising a characteristic set of features, com-mon to all pulses, such as rise times, polarities, magnitudes and times to maximumof mirror signals. These parameters are then defined as a function of the detectorposition, and a given pulse can be quickly assigned to a position. The drawbackof this approach is that superpositions of signals from nearby interactions resultin new values of such parameters. Thus it is most likely to be useful in detectorswhere multiple interactions in a segment or in adjacent segments are unlikely, as isthe case with double sided strip detectors (DSSD) [14]. The main advantage is thespeed of such analysis. These characteristics make the parametrization approachuseful in medical imaging applications, where a system must be able to cope withlarge rates of single events.

2.3.1 Timing

In any pulse shape analysis method that performs a comparison of a measured pulseshape to a basis, it is extremely important that the two are aligned in time. A poortime alignment results in incorrect fitting, as can be seen in fig. 2.3. The greatestdiscrepancy between the two unaligned signals is in the leading edge of the pulse,thus a least square or similar algorithm would find a basis pulse that minimizes thiswhile largely disregarding all other features in the waveform, such as the amplitudeor polarity of the image charges. In the example in fig. 2.3, the experimental signalis delayed with relation to the basis signals. One can see that the algorithm pickeda basis pulse with the lowest possible rise time to compensate.

The standard technique for timing in gamma spectroscopy is constant fractiondiscrimination, CFD. In the case of an arbitrary type of germanium detector it islikely that CFD would be necessary. In this study, however, a property specific tothe planar geometry was utilized. Consider a planar detector with unsegmentedcontacts where the dimension of the contacts is much greater than their separation.The relevant weighting field is then essentially uniform and thus the induced currenton either contact is constant until the charge is collected. This results in a linearlyincreasing charge signal. Considering that there are two charges in motion, thesignal consists of a sum of two straight lines (a line with a kink). A good timing

2.4. THE MATRIX METHOD 15

11 12 13

−200

−180

−160

−140

−120

−100

−80

−60

−40

−20

0

segment number

puls

e am

plitu

de, A

DC

cou

nts

measured pulsefitted pulse

Figure 2.3: PSA with erroneous timing. Here, the measured pulses are delayed incomparison to the basis pulses.

can thus be obtained by fitting a straight line to the first segment of such a linearpulse and finding its intersect with the baseline. The segmentation of one of thecontacts does not change the situation significantly other than splitting the signalbetween the segments. Thus the total signal from all segments of a planar detectorcan be used for timing, this is illustrated in fig. 2.4.

2.4 The matrix method

The task of deducing the position information from the pulse shapes becomes par-ticularly complex when more than one segment is hit. Provided that there are notat least 2 segments separating the target segments, one must take into account thesuperposition of transient signals from different interactions (in case of one-segmentseparation) and possibly of image signals with charge signals (in case of hits in ad-jacent segments). Such situations are far from uncommon and cannot be discardedwhile maintaining a good efficiency of the detector. In order to deal with suchsituations other than by omission of channels containing superimposed signals, themeasured signals must either be matched to a set of signals containing all possiblesuperpositions or the PSA algorithm must be able to return more than one signalfor a given event. The latter is the case for the matrix method [23].

Consider the interaction points of a γ ray and the corresponding energies de-


0 50 100 150 200 250−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

time, ns

Figure 2.4: Summation of signals. Total signal is shown as a thick line.

posited in those points to be the input to a system and the resulting pulse shapethe output. Both may be seen as vectors. The former, called vector x, is a list ofenergies deposited in all locations, where a few positive entries represent the inter-action points and their energies. The latter – a digitized signal waveform, s. Thesystem that relates the input to the output is then a matrix whose one dimensionmatches the number of the locations considered and the other - the length of thedigitized waveform. In short, such a system could be written as

Mx = s (2.2)

A matrix, M , that fulfills the requirement of this system is one where the columnsare the waveforms arising from interactions in a set of points in the detector. This isa linear equation system; in other words, it is a necessary assumption that the signalfrom two interactions is the sum of the signals from each constituent interaction onits own.

While equation 2.2 appears extremely simple, the matrix M contains a vastamount of information. It is perhaps still too early to claim whether a solutioncould be obtained in a realistic time for an on-line analysis in a germanium detec-tor system. It is, however, possible to reduce the amount of data greatly, whilepreserving enough information for a reasonable solution. A further complication isthat the solution x to eq. 2.2 may well include negative values – these are, however,

2.5. THE MATRIX METHOD: IN-DEPTH 17

unacceptable considering that the values are the deposited energies. Thus the sys-tem must be solved with a non-negativity condition. A further condition should bethat the number of non-zero elements in x is small - after all it is highly unlikelythat hundreds of points in a detector are struck simultaneously. The followingsection explores the matrix method in detail.

2.5 The matrix method: in-depth

We begin by creating a database of detector responses to single-point interactionsalong a grid of locations throughout the detector - this is the basis set and theselocations are the basis points (likewise - a basis grid). A part of this databasewhere only pulses from basis points located within one segment is a segment

basis set. For a given interaction it is convenient to collect the waveforms from allrelevant segments in a single vector called a meta-signal (fig. 2.5 for example). Inthis work, the meta-signals for the basis points were stored in the columns of thematrix M (eq. 2.2).

s = m1x1 +m2x2 + · · ·+mnxn =Mx (2.3)

All experimental signals will be fitted with linear combinations of the basissignals. Unlike many other methods, the result of the fit is not a single point buta set whose basis signals combine to give the best match to the measured signal.Clusters of close-lying basis points represent a single interaction whose energy is thesum of the energies at the basis points and the position is the average of the basispoint positions weighted with the assigned energies. This is possible because pulseshapes change nearly gradually as the interaction position is varied – i.e. withoutdiscontinuities. The procedure thus provides a significantly greater granularity thenthat given by the density of the basis points, see sections 4.2. It must be noted,however, that the identification of basis points belonging to the same cluster is farfrom trivial. This is, however, only an issue for points within the same segment -those in different segments are clearly separated since energy is measured by eachsegment is known. Furthermore, there may be more than one combination of basispulses which may give an adequate fit, the presence of noise making it impossibleto guarantee that any one is the correct solution.

2.5.1 Solving the matrix equation

To solve eq. 2.2, the matrixM must be inverted; alternatively, Gaussian eliminationmay be used. The result is the best fit in the least square sense. Generally, however,the resulting vector x will contain any real values including negative ones. Toavoid such non-physical solutions, the non-negative least square algorithm was used(NNLSA) [23, 24]. The algorithm is an iterative procedure that finds one positiveelement of x at a time and stops when a given tolerance is reached. Hence thecondition that the number of non-zero energy deposits must be relatively low is


1 2 3 4 5 6 7 8 9 10 11 12 13 14

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

Segment number

Figure 2.5: An experimental meta-signal (solid line) fit by the matrix algorithm(dots)

also fulfilled by the algorithm. The Matlab [25] implementation of the algorithmwas used. In general, the matrix M will have the number of columns equal tothe total number of basis points in the detector, while the number of rows will bethe number of samples of the pulse times the number of segments. The followingparameters help to reduce the size of the matrix.1

• number of waveform samples The waveform should be well represented,but it is unnecessary to have a sampling frequency that much exceeds thetypical frequency components in the signals. The length of the waveformneeds to be such that it covers the entire charge collection time as well assome extra samples before and after needed to accommodate pulses that maybe shifted by the timing algorithm.

• number of basis points used Only the basis sets for the segments thatactually measured non-zero energy need to be considered. It is practical tocalculate the basis sets for each segment and allow the algorithm to choosethe right matrix.

1As an example, in the 25-pixel 50 × 50 × 20 mm detector where each segment has a 5-by-

5-by-10 point grid, with 25 data points for each wave form, M would be a 6250-by-625 matrix.

Without data reduction, the solution time would be on the order of tens of seconds - unacceptably

slow even for off-line analysis.


• number of segments in a meta-signal The segments far away from theinteraction points only receive vanishing transient signals. It is sufficient touse only the immediate neighbours of the hit segment (an exception wouldbe a strip detector with narrow contacts where next-to-adjacent strips showsignificant singals).

• number of non-zero singular values The matrix M is in practice rank-deficient. Performing singular value decomposition, one finds that singularvalues decrease rapidly, and it has been shown in ref. [23], that setting themajority of singular values to zero while maintaining only a small number(typically 20), not only does not reduce the accuracy, but in fact improvesperformance in the presence of noise.

We begin with eq. 2.2. If the number of basis points in a segment is n and thenumber of samples in the entire meta-signal is l andM [l×n] is an l-times-n matrix,

M [l × n] x[n× 1] = s[l × 1] (2.4)

Performing singular value decomposition of M one obtains

U [l × l]W [l × n] V T [n× n] x[n× 1] = s[l × 1] (2.5)

whereW is a diagonal (non-square) matrix with the singular values placed along thediagonal ordered from greatest to smallest2. Setting all but the k greatest singularvalues to zero allows to discard large parts of the matrices, effectively reducingdimensions from n and l to k (note that the dimensions of the measured signal, s,and the energy deposit distribution in the detector, x, are unchanged). With thistruncation we obtain

U [l × k]W [k × k] V T [k × n] x[n× 1] = s[l × 1] (2.6)

W [k × k] V T [k × n] x[n× 1] = U−1[k × l] s[l × 1] (2.7)

(WV T )[k × n] x[n× 1] = U−1[k × l] s[l × 1] (2.8)

where U−1 is the pseudo-inverse of the matrix U . For a given basis matrix, M , thematrices in eq. 2.8 are constant regardless of the input data. This suggests thatthese matrices may be computed in advance and stored in the memory. In thiswork, the segment basis matrices, M , the matrices WV T and U−1 were calculatedfor every segment as well as for every combination of two segments for the detectorgeometries studied. Thus the solution requires a multiplication of U−1 by s followedby the solution of

2it is always possible to choose any order of singular values along the diagonal of W by

reordering rows and columns of U and V


(WV T )[k × n] x[n× 1] = (U−1s)[k × 1] (2.9)

using the non-negative least square algorithm. The events with more than twotriggered segments must be treated in a different way, since it is unrealistic to havethe matrices for all possible segment combinations in memory. While it may seemreasonable to perform the procedure in eq. 2.4-2.9 for a matrix M containing theentire detector basis, note that the total number of basis points n is not reduced inthe last equation. This creates a very large system for the NNLSA, and it was foundthat it is faster to assemble the basis matrix with the necessary segments only, dothe singular value decomposition and solve the system on an event-by-event basis.Fortunately, such events are quite rare and did not limit the overall speed of theanalysis.

2.5.2 The definition of the interaction points

The output of the matrix method is the vector x. Each element, xi is the assigneddeposited energy to the basis point i. The linear combination of all the basis signals,mi, given by

∑

i>0

ximi (2.10)

provides a fit to the measured pulse, see fig. 2.5. Fig. 2.6 shows the energy assignedto the basis points for the same event. An interaction point may be defined as theaverage position weighted with the assigned energy, in other words the centre ofmass of the points within one segment. This procedure works well if there indeedwas only one interaction in the segment. In the case of multiple interactions, it findsthe centre of mass of the interaction points. Nevertheless, this procedure was usedin the analysis presented in this work. Fortunately, the segments of planar detectorstend to be relatively small (compared to detectors such as those of AGATA), andthe events with multiple interactions per segment are in the minority.

In order to understand the difficulty of resolving multiple interactions in a seg-ment, let us consider the sets of basis points returned by the matrix algorithm.Figure 2.7 shows a reconstruction of a single interaction close to the lower left cor-ner of the segment. The basis points chosen for the fit include some at the oppositeedges of the segment. Note that this is actually a very good fit in terms of positionresolution, however if one was to attempt to find clusters of basis points and definemore than one interaction, it is likely that this event would be misinterpreted asa multiple interaction. It has been suggested that with an a priori knowledge ofthe number of points, provided by an additional algorithm, it may be possible toidentify multiple interaction points successfully [26]. This aspect, however, has notbeen investigated in the present work.


−20−10

010

20

−20−10

010

20

0

5

10

15

20

Figure 2.6: The basis points contributing to the fit in figure 2.5. The areas of thedots represent the assigned energy.

Figure 2.7: The basis points assigned to a single interaction are often spread outover a segment.

Chapter 3

Experimental applications

This chapter presents a summary of two experiments performed with the KTH 25-pixel planar detector described in section 1.2.1– Compton imaging and polarizationmeasurements. The full descriptions of these experiments are given in papers 1 and2. The pulse shape analysis method described in chapter 2 was used in both exper-iments. In each case, it was necessary to reconstruct events with two interactions– one Compton scattering and one photo-absorption. In practice, this implies theselection of events where two segments were triggered with the total energy equal tothat of the γ-ray of interest. Such selection does not, of course guarantee that thereare no more than two interactions – multiple interactions within each or either ofthe triggered segments are possible. It is however possible to suppress such eventsas well as to use the remaining ones in the analysis.

3.1 Compton Imaging

The concept of Compton imaging relies on the relationship between the energytransferred to an electron in a Compton scattering event to the angle between thedirections of the incoming and the outgoing photons. This relationship is given bythe formula

Ee− =

E2

γ

mec2(1− cos θ)

1 +Eγmec2

(1− cos θ)(3.1)

While the scattering angle θ is determined by the energy values, the remainingdegree of freedom is given by the momentum vector of the recoiling electron. Thiscannot be measured in most solid state detectors. Thus the direction to the sourceof the γ-ray can be constrained to the surface of a cone with the vertex in the firstscattering point, as in fig. 3.1. When the total energy of the γ ray is known theangle θ is obtained using only the first two scattering points, or – in case of only onescattering – the Compton and the photo absorption points. This can be utilizedif the photon can be assumed to be fully absorbed within the detector, or if the

23

24 CHAPTER 3. EXPERIMENTAL APPLICATIONS

overwhelming part of the radiation is monoenergetic, as is the case in most medicalapplications where a single isotope, e.g. 99mTc, is used. In cases of incompleteenergy collection, three scattering points are required in order to reconstruct acone [27].

Figure 3.1: The concept of Compton imaging and the 25-pixel planar detector.

The ability to reconstruct the interaction points of a γ ray to millimeter preci-sion is of great value for Compton imaging. Without of the pulse shape analysistechnique, the position resolution is constrained to the physical size of the detec-tors or detector elements. As a result, the detector systems must be very large.A relatively compact single germanium detector with a 3d position sensitivity iscapable of a 4π imaging. Alternatively, two, or more detectors can be used – someas scatterers and others as absorbers.

3.1.1 Imaging with the planar HPGe detector

The imaging capability as well as the attainable position resolution were testedusing the 25-fold segmented planar pixel detector. A 137Cs source emitting 662 keVphotons was used for the imaging tests. The source was placed 90 cm away fromthe detector, once in the normal direction to the cathode contact and once alongthe direction 45 off the normal to the cathode face.

In order to thoroughly test the PSA, it was decided to use only events withinteractions in neighboring pairs of segments, and only in the 9 inner segments. Inthis way, each interaction had a complete set of mirror signals available for PSA.At the same time, the segment signals generally are superpositions of transient and

3.1. COMPTON IMAGING 25

net charge signals. Such event selection resulted in two possible segment geometriesshown in fig. 3.2, with either 12 or 14 segments that were used for PSA. Theselection of the adjacent segments allows for a rather high imaging efficiency sinceit is unlikely that a scattered 662 keV photon interacts in a a segment far away fromthe scattering location in germanium. The efficiency, defined as the fraction of thephotons imaged out of those emitted into the solid angle of the segments used,was estimated at 5.6%. The attainable image resolution, however suffers from thischoice, since the proximity of the first and second interaction points limits theaccuracy in the definition of the cone axis.

Figure 3.2: The two cases of 2-segment events considered in this work. The seg-ments treated by PSA for each case are shown, the triggered segments are filled.

The image is reconstructed using filtered cone back-projection. The cone fromeach interaction is projected onto a sphere surrounding the detector. The inter-section of a cone with the sphere yields a circle. Figure 3.3 shows several coneprojections, here one hemisphere is mapped onto a square similarly to a typicalworld map. Note that many cone projections are only partially visible as linescrossing the entire map. This is a consequence of the detector geometry – therequirement of two triggered segments has a tendency to favour events where thephoton was scattered at an angle around 90, thus the reconstructed cone is verywide, often intersecting both hemispheres. The cones close to the poles (at thetop and bottom of the picture) are strongly distorted due to the map projection.A maximum likelihood filtering algorithm can be applied to obtain a smootherpicture.

3.1.2 Comparison with simulation

A GEANT3 simulation was compared to the experimental data analysis results.The Compton profile [28] was taken into account using the GEANT low energyCompton scattering (GLECS) package. It was found that for 662 keV photons,43% of all events where the full γ-ray energy is absorbed in the detector triggerexactly two adjacent segments and in 28% of those the interaction consists of two


Figure 3.3: Back-projections of cones onto a map of one hemisphere.

points, while in the remaining cases one or both segments contain more than oneinteraction point.

For a pair of interaction points, the photon cannot be tracked if the position ofthe source of the radiation is unknown, as is the case in imaging. The exceptions arethe events where the energy of one of the interactions is above the Compton edge,such a point cannot be the first interaction.1 Thus for many events two physicallypossible cones can be fitted to the two points. However, simulations have shownthat the segment that contains the larger energy deposit is the first one hit in80% of the 662 keV events, thus the segment with the greater energy deposit wasassumed to be the first interaction for all events, with the exception noted above.It is important to note that this probability is given strictly for the geometry of theplanar detector and the energy used in this experiment and should not be taken asa general statement. As only the events with a total absorption were selected, thegeometry of the absorber is essential, while the energy deposition in a Comptonscattering is strongly dependent on the γ-ray energy.

To accurately represent the experiment, interactions in pairs of adjacent seg-ments in the center of the detector were selected and any multiple interaction pointswithin the same segment were merged into a single point in the center of mass of theoriginal points. The resulting interaction positions were then randomly perturbedusing a normally distributed position uncertainty to match the resulting image tothat obtained in the experiment. The resolution of the image was found to matchthe experiment with a 1.5 mm (std. dev.) position uncertainty. A fixed 2 keVenergy resolution (FWHM) was applied to the simulated interactions. Figure 3.4

1provided that the γ-ray energy is known

3.1. COMPTON IMAGING 27

shows the images reconstructed from the experiment (a and b) and from the simu-lation (c and d). Between 4000 and 6000 events contributed to each image shown.The angular resolution of the image is approximately 30 (FWHM).

−90 −45 0 45 90

90

45

0

−45

90−90 −45 0 45 95

90

45

0

−45

−90

angl

e θ

(deg

rees

)

−90 −45 0 45 90

90

45

0

−45

−90

angle φ (degrees)−90 −45 0 45 90

90

45

0

−45

−90

a b

c d

Figure 3.4: Image reconstruction using filtered cone back-projection. a and b –simulated data, c and d – experimental data. The field of view corresponding toone hemisphere is shown.

3.1.3 Imaging resolution and efficiency

The obtained efficiency of 5.6% is high for a Compton imaging device, while theangular resolution is rather poor. As a comparison, in a similar experiment using acoaxial segmented detector [12] only non-neighboring segments were used, resultingin an efficiency of an order of magnitude lower (0.4%), while significantly increasingthe image resolution (to 5). The difference is expected due to the better definitionof the cone axis when the interaction points at a large separation are selected. Forthe planar detector used in this study it was found from the simulation that imagereconstruction where the hit segments are required to be separated by one segment


enhances the angular resolution of the image by a factor of 2.3 while lowering theefficiency by a factor of 3.5. In other words, a significantly higher angular resolutioncan be achieved at the expense of imaging efficiency. The combination of all possiblesegment combinations would of course maximize the efficiency, while also improvingthe image resolution as compared to data from adjacent segments only.

3.2 γ-ray polarimetry

Compton scattering can be used to measure the polarization of a beam of γ-rayphotons. A comprehensive review of the Compton polarimetry techniques can befound in [29]. The Compton scattering angle, θ, is defined as the angle betweenthe directions of the incoming and the outgoing photons. The azimuthal scatteringangle, φ, is defined as the angle between the polarization vector of the incomingphoton and the plane spanned by the incoming and outgoing photons’ momentumvectors. The differential cross section for Compton scattering into a solid angleelement dΩ is given by the Klein-Nishina formula:

dσ

dΩ=r2

0

2

E′2γE2γ

(E′γEγ

+EγE′γ− 2 sin2 θ cos2 φ) (3.2)

where r0 = e2

4πǫ0mec2is the classical electron radius. The scattering cross section is

enhanced in the direction perpendicular to the polarization vector of the incomingphotons, as visualized in fig. 3.5.

30

210

60

240

90

270

120

300

150

330

180 0

Figure 3.5: Azimuthal dependence of the Compton scattering cross section relativeto the polarization vector of the incoming photons. The photon energy is 288 keV.

3.2. γ-RAY POLARIMETRY 29

The sensitivity of Compton scattering to the polarization of photons of a givenenergy is expressed as the quantity known as the modulation fraction [30]

M(φ) =N(φ+ 90)−N(φ)

N(φ+ 90) +N(φ)(3.3)

where N(φ) is the intensity of scattered photons into the angle φ. The maximum ofM(φ) is reached when N(φ+ 90) is at its maximum and N(φ) is at its minimum.From fig. 3.5, it is evident that this occurs at φ = 0. Figure 3.6 shows the variationof the modulation fraction as a function of the scattering angle θ. We see that thepolarization sensitivity decreases with the photon energy. The maximum of M(φ)is at θ = 90 in the low-energy limit. With increasing energy it shifts towardssomewhat smaller angles. This feature can be utilized in the construction of apolarimeter – a detector system favoring a slightly forward-scattered events has anadvantage for higher energies.

0 20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

scattering angle, θ

Mod

ulat

ion

frac

tion

100 keV288 keV662 keV1300 keV

Figure 3.6: The dependence of the modulation fraction on the scattering angle forsome energies.

3.2.1 Polarization measurement

In order to test the performance of the 25-pixel detector as a polarimeter, a mea-surement using a 137Cs source was performed. The 661.7 keV γ-rays were collimated


towards a non-segmented coaxial HPGe detector. The Compton-scattered radia-tion from this detector is the source of polarized photons that were analyzed by the25-pixel detector. The schematic of the experiment is shown in fig. 3.7. 661.7 keVphotons scattered at 90 result in polarized 288.3 keV photons and a 373.4 keVenergy deposit in the scatterer. Due to the large size of the detectors, the possiblescattering angles for events with a single interaction in the coaxial detector varyfrom 85.5 to 95.0 resulting in energies measured in the coaxial detector between360.0 keV and 386.8 keV.

coaxial Ge

planarpixel Ge

70 cm

43 cm

Lead

Figure 3.7: The set-up for the polarization measurement.

Data was collected for all events where the two detectors triggered in coinci-dence. Those events where exactly two of the segments of the planar detector hadtriggered, with the total energy of 288 keV, were used for the polarization measure-ment. In the off-line analysis, a narrow energy gate was applied to the total energydeposit in both detectors. A further energy gate is then set for the energy in eitherone of the detectors. This ensures that the the photons scattered at angles close to90 in the coaxial detector are selected.

3.2.2 Analysis

Analysis was performed both with and without the use of PSA. Without PSA, anumber of scattering directions are defined by vectors connecting the centers of thepixels, with the majority of directions represented by several segment combinations.When PSA is used to reconstruct the interaction points, it becomes possible toperform kinematic event selection. The position resolution obtained in the imagingmeasurement (section 3.1) constrains the scattering angle, θ to a range of values.This range depends heavily on the separation of the interaction points. For points


closer than twice the position uncertainty, no constraint can be put on θ, thereforea minimum distance of 5 mm between the reconstructed interaction points was setfor events used. Furthermore, the Compton scattering formula was used to testwhether the obtained range of angles θ is consistent with the energies measuredin the two triggered segments. Events that do not fulfill this condition are likelyto consist of more than two interaction points and were rejected. Finally, angles θlying between 56 and 116 were selected in order to select the events lying in therange where the polarization sensitivity is greatest, see fig. 3.6.

The scattered intensities measured for different directions will vary greatly dueto the geometries of the two segments involved as well as the distance betweenthese segments. Assuming that all segments are identical in size, one finds thatsuch geometrical effects for a direction φ are identical to those at the angle φ+ 90.Therefore scattered intensities were normalized using [31]:

Inorm =I(φ)

I(φ+ 90)(3.4)

for scattering directions obtained using both the centers of the pixels as well as forthe PSA-reconstructed points.

The polarization of the 288 keV γ-rays analysed in the planar detector is notmaximal due to the method of their production. Since 662 keV γ-rays are scatteredat 90, according to fig. 3.6, the maximal obtainable modulation fraction is 0.578.The maximum modulation fraction for the 288 keV photons (assuming the optimalangle θ) is 0.840. It is therefore expected that the highest modulation fractionobtainable in the measurement is 0.485. Note that the modulation fraction isdefined for the intensity distribution, in contrast to the normalized intensity.

3.2.3 PSA versus pixels

Table 3.1 shows the modulation fractions obtained for the overall measurement aswell as for each segment separation separately when using pixel resolution and PSAreconstruction. Both methods tend towards the maximal value of 0.485, it is clearhowever that with the use of PSA it becomes possible to obtain significantly higherpolarization sensitivity for the cases of close-lying segments.

modulation fractionSegment separation pixel resolution PSA reconstruction

all 0.173 0.3500 0.156 0.3451 0.306 0.3512 0.417 0.427

Table 3.1: The measured modulation fractions

For the analysis without the use of PSA, the events were analyzed separatelydepending on the distance between the two segments. In the measurement with the


288-keV photons, in approximately 90% of the events, photons scatter between twoadjacent segments, 8% scatter across 1 segment and 2% across 2 segments (very fewscatter across 3 segments). This is due to both the increased absorption for segmentsfar apart, as well as the lower number of segment combinations contributing to agiven scattering direction. The normalized scattering intensity distributions foreach of these categories are shown in fig. 3.8.

0 45 90 135 180 225 270 315 3600

0.5

1

1.5

2

2.5

3

angle φ

Nor

mal

ized

inte

nsity

Figure 3.8: Normalized scattering intensity distribution for the data from adjacentsegments (stars and dotted line), one-segment separation (triangles and dashed line)and two-segment separation (circles and solid line). The maximum modulationfractions are 0.156, 0.306 and 0.417 respectively.

While the excellent statistics for the events including adjacent segments result invery small error bars and a very good fit to the theoretical curve, the modulation (asdefined in eq. 3.3) of this distribution is significantly lower than for higher segmentseparations. This effect is expected because for adjacent segments, a wide range ofboth angles θ and φ are possible. A wide distribution in θ results in the inclusionof events for which the sensitivity to polarization is low (see fig. 3.6), while a poordefinition of φ causes an averaging effect for many possible scattering directions,thus reducing the modulation at its maxima and minima. The most pronouncedmodulation is found for segment pairs separated by two other segments, becausethen the effects described above are minimized.

Using PSA-reconstructed interaction locations, allows the selection of events


with the greatest sensitivity to polarization even in cases of close-lying interactions,as well as to reduce the number of events with multiple scattering points. The resultis shown in fig. 3.9. Approximately 20% of the events were kept in this case – themajority of the events separated by at least one segment as well as approximatelyone in eight of the adjacent-segment events. This possibility is important for low-energy γ-rays in particular. When the mean free path of the photons becomestoo short to obtain a significant number of events with separated segments, a highpolarization sensitivity can none the less be obtained using the adjacent segments.

0 50 100 150 200 250 300 3500

0.5

1

1.5

2

2.5

3

angle φ

Nor

mal

ized

inte

nsity

Figure 3.9: Normalized scattering intensities for the PSA-processed data for allpixel combinations. The maximum modulation fraction is 0.35.

Chapter 4

DESPEC planar detectors

The DESPEC array at the FAIR facility, GSI, will consist of an active stopper,namely a silicon implantation detector, surrounded by a germanium tracking ar-ray for γ-ray spectroscopy and/or neutron detectors. The information from theimplantation detector will be correlated with the γ-ray spectrometer. One of theproposed germanium arrays consists of 24 detector modules each housing three pla-nar crystals in a single cryostat. The size of the crystals is 72×72×20 mm. Thearrangement of the three crystals is shown in fig. 4.1.

72

6

20zx

y

Figure 4.1: The 3-crystal DESPEC detector module. Dimensions are given in mm.

Two types of segmented detectors are considered – a pixel detector, similar tothe detector described in section 3.1, and a double-sided orthogonal strip detector.In this work the performance of these two detector geometries from the point ofview of pulse shape analysis is examined. For the given size of the crystal, and anequal number of readout channels, the two possibilities are: 16 pixels whose size is17×17 mm and an 8-by-8 strip segmentation with the strip pitch of 8.5 mm. Forboth cases, 2-mm guard rings are assumed around the segmented contacts. The

35

36 CHAPTER 4. DESPEC PLANAR DETECTORS

geometries are illustrated in figure 4.2.

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4.1 Simulation

In order to test the performance of pulse shape analysis for the two types of planardetectors, a GEANT4 simulation of photons interacting with the detector was used.Photons of 200, 600, 1200 and 2000 keV were generated, and the interaction pointswithin the array of three germanium crystals were determined. The resulting pulseshapes for each detector crystal and segment were determined for the pixel andstrip segmentations as described in section 2.2. These were stored and, with addedwhite noise with the maximum amplitude corresponding to 2.5 keV, treated asexperimental signals. Additionally, the energies in each segment were determinedas the sum of energies of all interaction points in the space subtended by thesegment.

The crossing strips define a larger number of voxels than possible with the samenumber of pixels. In order to avoid confusion let us define the segment – namelythe volume which contributes to the signal on a given contact; and voxel – thesmallest volume element defined by simple readout (without PSA). The reasonthis distinction is convenient is that a segment is a unit of the electronic readout– something that yields a pulse shape, an energy and timing measurement. Thevoxel is the unit of physical spatial resolution of the detector. In the detectors withonly one segmented contact, such as pixel planar detectors or coaxial detectors,segments are identical to voxels. When it comes to the strip detector however, thevoxels are formed by the intersection of the segments.

4.1. SIMULATION 37

The simulated pulses were analyzed using the matrix method described in sec-tion 2.4. The total number of segments is 16 for the strip detector and 17 (includingthe non-segmented one) for the pixel detector. The sampling frequency was reducedto 50 MHz and 12 pulse samples per segment were used, resulting in 192 and 204elements per meta-pulse (l in eq. 2.4-2.9). The matrices described in section 2.4were prepared for single and double voxel interactions.

The segments far from the interaction points were not discarded, resulting inmeta-signals of a constant length. The data reduction would not be significant inthe case of the detectors in question. Consider double-voxel interactions: for thepixel detector, if all adjacent segments are used for reconstruction as well as thenon-segmented contact, depending on the relative positions of the two hit segments,between 7 and 15 out of 17 segments would be required. For the strip detector,the next-to-adjacent strips have very significant transient signals that are desirablefor PSA. This means that depending on the relative position of the hit voxels thenumber of strips to be kept would be between 5 and 16 out of 16. The events thattruly bottleneck the calculation are the ones with more than two voxels triggered,for which there are no pre-calculated matrices. For these, the number of necessarysegments is even larger. A further reason to avoid segment selection in multi-voxelevents is that the selection would need to be performed for each event, preventingthe use of pre-computed matrices; therefore it was judged that a segment selectionwould impair the algorithm more than benefit it.

4.1.1 Segments and voxels

The strip geometry presents a new difficulty to the PSA algorithm. The energiesread out on strips are the sums of energies in the voxels they contain. It is easy torealize that for an arbitrary distribution of interaction points in a strip detector, itwill not be possible to uniquely determine the voxels that contribute. In the case ofthe 8-by-8 strip DESPEC detector, the ambition is to treat the 64 strip intersectionsas voxels, the read-out, however, provides only 16 channels. Furthermore, thecondition that the total measured energy on one side should be equal to that onthe other side imposes an additional constraint. Thus one is left with 15 equationsand 64 unknowns. This ambiguity is illustrated in fig. 4.3. Note that PSA cannothelp to distinguish the 3 possibilities shown, because each strip has a vanishingposition sensitivity in the dimension parallel to it.

The following four categories of voxel configurations are sufficient to describeall possible events. See fig. 4.4 for relevant examples.

1. Only one strip has triggered on either one side of the detector. These areuniquely resolved - one simply uses the energy of the strips on the oppositeside to define the voxel energies. Single interaction events, of course, also fallinto this category.

2. Each strip triggered only collects energy from one voxel. An example is wherevoxels along the detector’s diagonal are hit. In this case one can match the


energies from every strip from one side to the energies on the other side.Pairs of strips with equal energies define a voxel. In some cases however thisprocedure will misidentify voxels if sums of energies are equal to within amargin given by the energy resolution.

3. Non-resolvable events – at least one voxel’s energy is summed in strips fromboth sides.

4. A combination of 1, 2 and 3.

8

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Figure 4.3: Example of an event where the voxels cannot be uniquely identified.Here the horizontal strips measure energies 5 and 15 and the vertical ones measure8 and 12. Three of the possible energy assignments are shown.

The PSA algorithm written for the strip detector resolves each situation in thefollowing way.

• Case 1 is solved directly.

• Cases 2 or 4 – any equal energies from the opposite strips are found one pairat a time, until case 1 or 3 is obtained.

• Case 3 – the under-determined system of equations for the measured energiesis set up and solved using the non-negative least squares algorithm.

The solution for the non-unique cases (case 3) attempts to find the configurationwith the smallest number of voxels that reproduce the strip energies. Even with thisconstraint, there is more than one possibility as shown in fig. 4.3. For the simulateddata used in the study, the correct identification took place in approximately 50% ofcategory 3 events. Due to this, voxels were incorrectly identified in approximately3% of all events for the 600, 1200 and 2000 keV γ-rays and less than 1% for the200 keV γ-rays. Note that an incorrect identification implies that the interactionpoints are distributed to the wrong voxels by the PSA algorithm, and typically, allreconstructed interactions in the detector in question will be false.

All interaction points in a voxel were merged and replaced by a centre-of-masspoint by the PSA algorithm. On average, 18% of the events in the strip detectorand 30% in the pixel detector had some merged points in at least one of the threedetector crystals due to this effect.

4.2. CHOICE OF THE BASIS GRID 39

case 1 case 2 case 3 case 4

Figure 4.4: Examples of the four possible voxel combination categories. Note thatin the right-most picture there is a combination of cases 1 and 2 in the upper leftcorner and case 3 in the bottom right corner.

4.2 Choice of the basis grid

A number of different basis grids using different densities of basis points were tested.The number of basis points defines the size of the matrices used in the matrixmethod (n in eq. 2.4- 2.9), and must therefore be optimized to make the CPUtimes realistic.

An important requirement for the grid is that the signals from the locationsclose to the edges of the segments be represented, because these locations producesignals with the most extreme features. Consider a transient signal in a segment andits dependence on the position of the interaction points in a neighboring segment(target segment). The greatest transient charge is obtained for an interaction ata point just outside of the segment, interactions in all more distant locations inthe target segment will create smaller transient charges (see fig. 4.5). Thus a basisgrid can only describe the points that are not closer to the edges than the grid’smost extreme points. On the other hand, any point in the target segment generatesa transient signal in a neighboring segment that is smaller than from a point atthe edge closest to the segment and greater than that from the opposite edge. Ifone was only interested in fitting the amplitude of the mirror signal it would besufficient to simply use a linear combination of signals from these two extremepoints. Moreover, if the amplitude changes linearly with the x-coordinate, thislinear combination accurately reproduces the position of the interaction (in x). Asseen in fig. 4.5, the dependence is not linear, it could however be approximated bytwo line segments if the two extreme points and the point in the centre are used.

The amplitude of one mirror signal is only one example of a parameter thatneeds to be fit. As another example is the dependence of the slope of the lead-ing edge of the net charge signal as a function of the z-coordinate. The pointsclose to the contacts represent the most extreme cases. All features need to be


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4.3. PSA RESULTS 41

accurately represented simultaneously, therefore the result of the application of thePSA algorithm is a compromise between all such features that minimizes the over-all difference between the measured and the fitted signals. The results of tests withvarious grids of basis points have indicated that it may be that a grid with mostlyedge and other extreme points can be adequate, provided that the features of thepulse shapes change gradually from one extreme to the other.

The grids that were tested are in the following described as nx×ny×nz where nx,ny and nz are the numbers of segment basis points along the respective coordinate.Due to the symmetry of the detectors, nx = ny was always chosen. All grids usedare irregular – the density of the points is increased in the proximity of the segmentedges in order to represent the rapidly changing pulse shapes. The grids used arelisted in table 4.1

Strip detector pixel detector(64 voxels) (16 voxels)

3×3×7 5×5×73×3×14 5×5×145×5×14 9×9×147×7×14 13×13×14

Table 4.1: The basis grids used in this work (nx × ny × nz). Grids with overallsimilar total numbers of points were chosen. As only odd nx and ny were used, thetotal number of points cannot be chosen exactly equal.

4.3 PSA results

The detailed presentation of the PSA results can be found in paper 3. Here, thekey results are summarized in table 4.2. The mean errors in reconstructed inter-action positions have been determined for each of the basis grids for both detectorgeometries. For each reconstructed point the distance to the original point wasdetermined. In the cases where two or more of the original points ware merged intoa single reconstructed point, the position error was defined as the energy-weightedaverage distance from the original points to the reconstructed one. The positionerrors were studied separately for the events both with and without merging, aswell as for events with one or more than one voxel hit per detector.

It was found that the position errors in the pixel detector were between a factorof 1.5 and 2 greater than in the strip detector for all types of events. This differencewas attributed primarily to the difference in the lateral voxel size. In order to morethoroughly explore this, a comparison was made to identical PSA algorithm appliedto the 25-pixel detector described in chapter 3.1. Table 4.2 shows that the positionresolution of the 25-pixel detector is closer to that of the DESPEC strip detector.The pixels of this detector are only slightly larger than the voxels of the stripdetector, and so are the position errors. Finally, a comparison with a detector


where a 10×10 mm granularity is reached using a geometry with 5+5 orthogonalstrips was also performed. The results here are similar, although surprisingly, theerrors are somewhat smaller for the strip detector. While the exact reason for thishas not been determined, there are two major differences. The number of segments,and thus of the pulse shapes, is greater for the pixel detector. It is possible thatthe inclusion of pulses that contain little or no information adversely affect theanalysis. It should be noted, however, that this effect, if it does exist, would beless important for a detector with a smaller number of pixels, such as the 16-pixelDESPEC detector. The other difference is that in the case of the strip detector, thecontacts on both sides contribute information to the PSA, while in a pixel detector,the non-segmented contact contributes little.

Detector DESPEC DESPEC16-strip 10-strip 25-pixel 16-pixel

voxel size (x×y) 8.5×8.5 mm 10×10 mm 10×10 mm 17×17 mmbasis grid 5×5×14 7×7×15 7×7×15 7×7×14single interactions 1.14 1.15 1.47 2.13multiple interactions 2.20 2.17 2.41 3.92single, merged 2.11 2.06 2.30 3.23multiple, merged 2.58 2.53 2.65 4.24single, non-merged 0.99 0.91 1.24 1.79multiple, non-merged 2.05 1.93 2.24 3.68

Table 4.2: Comparison of the DESPEC pixel and strip detectors to the 10-stripand 25-pixel detectors. Reconstructed position errors in mm are presented forsingle and multiple-voxel interactions. Average errors as well as errors for mergedand non-merged interactions are presented.

For both DESPEC detectors, the position errors were found to be as much as afactor of 2 greater for the events where multiple voxels were hit. A detailed studyof the errors on the event-by-event basis revealed the cause for this difference. Verylarge errors occur for low-energy interactions that are accompanied by one or moremuch bigger energy deposits in other voxels. This effect is expected because thetransient pulses from larger interaction may be greater than a net charge pulse froma low-energy interaction, and thus in superpositions of the pulse shapes from sucha pair of interactions, the features originating from the smaller one are obscured bythe larger one.

The choice of the density of the basis grid was found to have only a small effecton the position resolution. This is encouraging since a very large reduction in thedimension of the matrix equation 2.9 becomes possible, thus greatly decreasingthe solution time and the memory requirement. In addition to the study of thethree-dimensional position errors, the mean errors in position in each coordinatewere determined. The result may be considered surprising. In general, the changesin the grid density in x-y coordinates and in the z coordinate do not affect the

4.4. TRACKING 43

position errors in only those dimension but in all three dimensions simultaneously.This illustrates the complexity of the problem – there is a balance between allfeatures in the pulse shapes while performing the analysis.

4.4 Tracking

The PSA-reconstructed data was used as the input to a tracking algorithm designedfor DESPEC planar detectors [32]. The goal of tracking is ultimately to producea spectrum that is as free of background as possible, in other words to reject asmuch of the Compton continuum as possible as well as to disentangle multipleinteractions. Tracking may also enable the reconstruction of the total energy forthe events where an incomplete collection took place within the detector volume.A figure of merit defines whether a track is accepted.

The results of tracking are presented here as the peak-to-total ratio (P/T) asa function the tracking efficiency, defined as the number of full energy events thatare accepted by the algorithm. The tracking efficiency has been normalized on thephoto-peak efficiency of the detectors. Figures 4.6 show the tracking results forthe two detectors. In figure 4.7, the tracking results for the 2 MeV photons arecompared to simulations where tracking is applied to interaction points that havenot been processed by PSA, but instead a position uncertainty was added. In thesesimulations, it was assumed that interaction points merge if the distance betweenpoints is less than twice the position uncertainty. While this is not exactly thecase in the PSA-processed data, where merging is determined by the segmentationpattern, the results are very close if the merging distance is set to be just below thesegmentation size. Finally, tracking was applied to non-merged events separately.As can be seen in figure 4.7, merging has a minimal effect on the efficiency andpeak-to-total characteristics of the reconstructed γ-rays. This can be understoodconsidering that the close-lying points that are likely to be merged are common forlow-energy photons or the ends of the tracks of in case of photons with high energy.In either case, as such interaction points belong to a single photon, their energydeposits should ultimately be summed during tracking, and that this summingoccurs during PSA instead, does not affect the overall result.

4.5 Conclusion

The pulse shape analysis and tracking simulations show that the planar double-sided strip detector has a clear advantage over the pixelated geometry with thesame number of readout channels and the same sensitive volume. The effectivevoxel size, corresponding to the physical granularity of the detector, is a factor 2higher in the case of the strip detector, and the position sensitivity appears to scalewith this parameter. In other words a 64-pixel detector would be expected to havea position resolution similar to that of the 8×8 DSSD detector. Such an increasein the number of the readout channels implies not only a great increase in the cost


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Figure 4.6: Tracking results for the strip and pixel geometries for the γ-ray energiesused.

4.5. CONCLUSION 45

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of a detector system, but also an equal increase in the data to be processed by thePSA algorithm.

Considering the equal number of channels condition, a further advantage of thestrip detector’s smaller voxels is the lower probability of interaction point mergingthat takes place for the interactions inside one voxel. A serious drawback of theDSSD encountered in this work is the ambiguity in the voxel identification forthe events with many interaction points. For the events used here this affectedapproximately 3% of events, however, the problem may become more severe inexperiments with high γ-ray multiplicities, where a much larger fraction of eventsmay not be resolvable even to the precision of the physical granularity.

A further drawback for the strip detector in the present algorithm implemen-tation is that the number of pre-computed matrices (see. eq. 2.9) required forsingle-voxel events is equal to the number of voxels, n. For two-voxel events thenumber of combinations, and hence the number of matrices, is n(n − 1)/2. Thisamounts to 16 + 120 = 136 matrices for the pixel detector and 64 + 2016 = 2080matrices for the strip detector that need to be stored in the memory.1 On the otherhand, the solution times have generally been shorter for the strip detector due to

1This occupied approximately 1 GB of memory in case of the densest grid.


the lower number of basis points per matrix. This effect also allows for a faster com-putation in the cases of complex interactions where no pre-computed matrices areavailable. If the number of basis points is further minimized, it is possible that thegeneral matrix calculation can be performed as fast as that using the pre-computedmatrices without the great memory requirement. This would also be the case ina detector with a segmentation much finer than the ones studied here. As a finalconclusion, it can be said that the increased voxel size comes at the cost of positionresolution, the number of basis points per voxel cost in the processing time and thenumber of voxels cost in the amount of necessary memory.

Appendix A

Digital data acquisition

In order to perform PSA, the signals must be available in a digital form to beprocessed in a computer or a digital signal processor (DSP). Furthermore, theenergy of the pulses must be measured and the trigger condition defined. For theexperimental part of this work, a VME-based data acquisition system built byStruck Innovative Systems [33] was used. The preamplifier signals were connectedto flash ADCs (analogue-to-digital converters) that digitized the signal with 14-bitprecision at 100 MHz. In other words the digitized signal is given by values between0 and 16384 every 10 ns. The dynamic range of the ADCs is 2 V (i.e. ADC value16384 is 2 V above the ADC value 0).

The digitized signal arrives at a field programmable gate array (FPGA). Herethe signal is passed through two filters, one defining a trigger, the other measuringthe energy. Both of these filtered signals are available continuously with a shortdelay. There is also a delayed copy of the raw ADC waveform which is sampledwhen the trigger condition is met, ensuring that the sampled pulses contain databeginning before the trigger was created.

The energy filter is based on the moving window deconvolution (MWD) method.The signal is integrated and differentiated, resulting in a trapezoidal signal. Theintegration constant defines the time from the beginning of the trapezoid to the flattop (peaking time), and the differentiation constant - the time from the beginningof the trapezoid to the end of the flat top (shaping time). The height of this signalrelative to its baseline gives the energy of the pulse (this can be sampled in apredefined position or at the maximum value at the flat top), see fig A.1.

The situation is complicated somewhat since on the scale of the integration timenecessary, the exponential decay of the preamplifier pulses (approx. 50µs) is notnegligible. An additional term is added to the filter to correct for this - the failureto do so results in a trapezoid with a suppressed top and a trailing edge extendingbelow the baseline, creating incorrect energy readings for pulses arriving before thebaseline has a chance to recover. This is reminiscent of the pole/zero correctionin analogue linear amplifiers. A thorough study of the MWD algorithm has been

47

48 APPENDIX A. DIGITAL DATA ACQUISITION

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done by M. Lauer [34].A faster trapezoidal filter is used to define the trigger. This gives a much

rougher measure of the energy, however the decision whether to trigger or not canbe made much sooner. The trigger information from all ADCs is sent to a commonclock module that initiates a simultaneous trigger in all modules, thus sampling thesignals from all or a selection of segments (or detectors) simultaneously.

Appendix B

Glossary

Net charge signal the signal measured at a contact that collects the electrons orthe holes created in a γ-ray interaction and shows a non-zero charge after thecollection

Transient charge signal (also: mirror charge signal) a signal measured at a con-tact that does not collect any charge – transient charges vanish after thecharge is collected

Contact the conductive area on the surface of a detector where a signal can bemeasured

Segment the volume inside a detector from where the charges are collected ontothe same contact

Pixel a segment in a pixeleted detector (pixels tend to have square correspondingcontacts)

Strip a segment in a strip detector (the corresponding contact is long and narrow)

Voxel the volume inside a detector from which charges are collected onto a givencontact on one side and onto a given contact on the other side (identicalto pixel in a pixelated detector and to the intersection of strips in a stripdetector)

PSA pulse shape analysis – any of a number of techniques for determining thepositions of interactions of a γ-ray in a detector

γ-ray tracking a technique for finding a probable path traversed by a γ-ray basedon the interaction locations and their respective energies provided by PSA

DSSD double-sided strip detector

NNLSA non-negative least squares algorithm

49

50 APPENDIX B. GLOSSARY

Detector basis a set of pulse shapes calculated for a number of points that ade-quately represent the detector response

Segment basis a subset of a detector basis that adequately represents detectorresponse for interactions within one segment only

Basis point one point from a basis

Basis grid pattern according to which the basis points forming a basis are chosen

Meta-signal a one dimensional array where the digitized pulses from several con-tacts are stored one after the other

Acknowledgements

This work was supported by the Göran Gustafsson Foundation, the Swedish Re-search Council and the European Commission under contracts No RII3-CT-2004-506065 and RII3-CT-2004-506078. The author would like to thank B. Cederwallfor supervision and help with this work as well as A. Johnson and T. Bäck forproofreading this thesis.

51

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