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ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΑΤΡΩΝ

ΤΜΗΜΑ ΙΑΤΡΙΚΗΣ ΤΜΗΜΑ ΦΥΣΙΚΗΣ

ΔΙΑΤΜΗΜΑΤΙΚΟ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ

ΣΤΗΝ ΙΑΤΡΙΚΗ ΦΥΣΙΚΗ

ΑΝΆΠΤΥΞΗ ΜΕΘΌΔΩΝ ΠΡΟΣΟΜΟΊΩΣΗΣ

ΜΕ ΤΕΧΝΙΚΈΣ MONTE CARLO

ΔΙΑΔΙΚΑΣΙΏΝ ΠΑΡΑΓΩΓΉΣ ΣΉΜΑΤΟΣ ΣΕ ΦΩΤΟΑΓΏΓΙΜΑ

ΥΛΙΚΑ ΆΜΕΣΩΝ ΑΝΙΧΝΕΥΤΏΝ ΕΝΕΡΓΟΎ ΜΉΤΡΑΣ

ΣΤΗΝ ΨΗΦΙΑΚΉ ΜΑΣΤΟΓΡΑΦΊΑ

ΣΑΚΕΛΛΑΡΗΣ Β. ΤΑΞΙΑΡΧΗΣ

ΔΙΔΑΚΤΟΡΙΚΗ ΔΙΑΤΡΙΒΗ

ΠΑΤΡΑ 2008

UNIVERSITY OF PATRAS

SCHOOL OF MEDICINE DEPARTMENT OF PHYSICS

INTERDEPARTMENTAL PROGRAM OF POSTGRADUATE

STUDIES IN MEDICAL PHYSICS

DEVELOPMENT OF A MONTE CARLO SIMULATION MODEL

OF THE SIGNAL FORMATION PROCESSES

INSIDE PHOTOCONDUCTING MATERIALS

FOR ACTIVE MATRIX FLAT PANEL DIRECT DETECTORS

IN DIGITAL MAMMOGRAPHY

SAKELLARIS V. TAXIARCHIS

DOCTORATE THESIS

PATRAS 2008

THREE MEMBERS ADVISORY COMMITTEE

Professor George Panayiotakis, Main Supervisor

Professor George Nikiforides, Member of the Advisory Committee

Professor George Tzanakos, Member of the Advisory Committee

SEVEN MEMBERS EXAMINING COMMITTEE

Professor George Panayiotakis, Main Supervisor

Professor George Nikiforides, Member of the Advisory Committee

Professor George Tzanakos, Member of the Advisory Committee

Professor Nikolaos Pallikarakis, Member of the Examination Committee

Professor Spiridonas Fotopoulos, Member of the Examination Committee

Associate Professor Alexandros Vradis, Member of the Examination Committee

Assistant Professor Eleni Costaridou, Member of the Examination Committee

Acknowledgments

ACKNOWLEDGMENTS

The deepest gratitude to my supervisor Professor George Panayiotakis for offering me the

opportunity to make this PhD and for his continuous support and guidance during all these years.

I am truly grateful and indebted to Dr. George Spyrou, for his precious help, guiding, ideas

and advices he gave me during this work. His contribution has been essential.

I would like to thank Professor George Tzanakos for his participation and useful

discussions we had all these years.

I would also like to thank Professor George Nikiforides for his co-operation and his support

during this PhD thesis.

I am also grateful to Associate Professor Eleni Costaridou for the valuable help she offered

by providing me with important bibliography relevant to this PhD thesis.

Also, I need to thank Associate Professor Alexandros Vradis for the useful meetings we

had and the important advices he gave me.

I would like to express my gratitude to all my colleagues in the Department of Medical

Physics as well, for the helpful discussions we had but also for the moments we lived together

throughout all these years.

Finally, I would like to thank the State Scholarship Foundation of Greece (ΙΚΥ) that

supported this research work by a grant.

TABLE OF CONTENTS

TABLE OF CONTENTS

Chapter 1. Introduction.............................................................................................................11.1. Introduction........................................................................................................................21.2. Thesis.................................................................................................................................41.3. Thesis layout......................................................................................................................61.4. Publications........................................................................................................................61.5. Financial support...............................................................................................................7

Chapter 2. Mammography........................................................................................................82.1. Introduction........................................................................................................................92.2. Mammographic Equipment...............................................................................................9

2.2.1. X-ray tube.................................................................................................................102.2.2. Compression device..................................................................................................112.2.3. Antiscatter grid.........................................................................................................112.2.4. Image receptors.........................................................................................................12

Chapter 3. Mammographic Detectors....................................................................................133.1. Introduction......................................................................................................................143.2. Phosphors.........................................................................................................................143.3. Film-Screen systems (Analog detectors).........................................................................143.4. Digital Detectors..............................................................................................................15

3.4.1. Phosphor-CCD systems............................................................................................153.4.1.1. CCD devices......................................................................................................153.4.1.2. Optical coupling of a phosphor to a CCD.........................................................163.4.1.3. Slot scanned digital mammography...................................................................18

3.4.2. Photostimulable phosphors (Computed radiography systems).................................183.4.3. Active matrix flat panel imagers (AMFPI)...............................................................20

3.4.3.1. Indirect detection...............................................................................................203.4.3.2. Direct detection.................................................................................................21

3.5. Digital versus Film-Screen Mammography.....................................................................24Chapter 4. X-Ray Photoconductors........................................................................................26

4.1. Introduction......................................................................................................................274.2. The ideal x-ray photoconductor.......................................................................................274.3. Crystalline, amorphous and polycrystalline solids..........................................................284.4. X-ray photoconductors....................................................................................................29

4.4.1. Amorphous Selenium (a-Se).....................................................................................304.4.2. Polycrystalline Cadmium Telluride (poly-CdTe).....................................................314.4.3. Polycrystalline Cadmium Zinc Telluride (poly-CdZnTe)........................................314.4.4. Polycrystalline Lead Oxide (poly-PbO)...................................................................324.4.5. Polycrystalline Mercuric Iodide (poly-HgI2)............................................................324.4.6. Polycrystalline Lead Iodide (poly-PbI2)...................................................................33

4.5. Table of material properties.............................................................................................34Chapter 5. Physics Of Image Formation................................................................................35

5.1. Introduction......................................................................................................................365.2. X-ray – matter interactions..............................................................................................36

5.2.1. Coherent (Rayleigh) scattering.................................................................................365.2.2. Incoherent (Compton) scattering..............................................................................37

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TABLE OF CONTENTS

5.2.3. Photoelectric absorption...........................................................................................375.3. Atomic deexcitation.........................................................................................................385.4. Electron interactions........................................................................................................40

5.4.1. Elastic scattering.......................................................................................................405.4.2. Inelastic scattering....................................................................................................42

5.4.2.1. The development of a theory for inelastic collisions.........................................425.4.2.2. Recoil energy.....................................................................................................435.4.2.3. Bethe’s theory revisited.....................................................................................445.4.2.4. Generalized Oscillator Strength -Optical Oscillator Strength..........................455.4.2.5. Bethe surface- Bethe sum rule...........................................................................475.4.2.6. The differential inelastic scattering cross section.............................................475.4.2.7. Secondary electron emission (δ-rays)................................................................49

5.4.3. Collective description of electron interactions.........................................................495.5. Charge carrier transport inside a-Se................................................................................50

5.5.1. Geminate (Onsager) recombination..........................................................................525.5.2. Columnar recombination..........................................................................................53

5.5.2.1. Absence of electric field.....................................................................................545.5.2.2. Field parallel to the column..............................................................................555.5.2.3. Field perpendicular to the column....................................................................555.5.2.4. Field at an angle φ with the track......................................................................55

Chapter 6. Monte Carlo Simulation........................................................................................566.1. Introduction......................................................................................................................576.2. Random numbers-Random variables...............................................................................586.3. Probability Distribution Functions - Cumulative Distribution Functions.......................596.4. Sampling techniques........................................................................................................60

6.4.1. The Inversion Method...............................................................................................616.4.2. The Rejection Method..............................................................................................62

Chapter 7. Primary Electron Generation Model...................................................................637.1. Introduction......................................................................................................................647.2. Electron from Incoherent Scattering................................................................................647.3. Photoelectric absorption..................................................................................................65

7.3.1. Photoelectric absorption from a molecule................................................................657.3.2. Production of photoelectron.....................................................................................66

7.4. Atomic deexcitation.........................................................................................................687.4.1. K and L shell deexcitation........................................................................................687.4.2. Simulated atomic transitions.....................................................................................697.4.3. Energies and directions of fluorescent photons, Auger and CK electrons...............71

7.5. Model limitations.............................................................................................................71Chapter 8. Primary Electron Generation: Results & Discussion.........................................73

8.1. Introduction......................................................................................................................748.2. Energy distributions.........................................................................................................75

8.2.1. Fluorescent photons..................................................................................................758.2.2. Escaping photons......................................................................................................758.2.3. Primary electrons......................................................................................................75

8.2.3.1. Monoenergetic case...........................................................................................778.2.3.2. Polyenergetic case.............................................................................................78

8.3. Angular distributions of primary electrons......................................................................80

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TABLE OF CONTENTS

8.3.1. Azimuthal distributions............................................................................................808.3.2. Polar distributions.....................................................................................................83

8.4. Spatial distributions of primary electrons........................................................................858.4.1. Monoenergetic case..................................................................................................868.4.2. Polyenergetic case....................................................................................................94

8.5. Arithmetics of photons and primary electrons.................................................................958.5.1. Arithmetics of escaping primary photons.................................................................968.5.2. Arithmetics of fluorescent photons produced...........................................................968.5.3. Arithmetics of escaping fluorescent photons............................................................988.5.4. Arithmetics of escaping primary and fluorescent photons.......................................988.5.5. Arithmetics of primary electrons produced............................................................1008.5.6. Summary tables......................................................................................................101

Chapter 9. A Preliminary Study On Final Signal Formation In a-Se...............................1039.1. Introduction....................................................................................................................1049.2. Mathematical formulation.............................................................................................1049.3. Results and Discussion..................................................................................................106

9.3.1. Energy distribution of primary electrons on top electrode.....................................1069.3.2. Time distribution of primary electrons on top electrode........................................1069.3.3. Spatial distribution of primary electrons on top electrode......................................1079.3.4. Angular distributions of primary electrons on top electrode.................…………109

Chapter 10. Electric Field Considerations In a-Se..............................................................11010.1. Introduction..................................................................................................................11110.2. Boundary conditions....................................................................................................11110.3. Calculation of the electric potential distribution.........................................................113

Chapter 11. Model Formulation For Electron Interactions In a-Se..................................11711.1. Introduction..................................................................................................................11811.2. Electron free path length..............................................................................................11811.3. Decision on the type of electron interaction................................................................11911.4. Elastic scattering..........................................................................................................119

11.4.1. Differential cross section......................................................................................11911.4.2. Elastic scattering cross section (σel)......................................................................121

11.5. Inelastic scattering.......................................................................................................12211.5.1. Inelastic scattering with inner shells (K and L shells)..........................................12211.5.2. Inelastic scattering with outer shells.....................................................................124

Chapter 12. General Discussion, Conclusions & Future Work..........................................12812.1. General discussion.......................................................................................................12912.2. Conclusions and future work.......................................................................................133

References................................................................................................................................136Abstract...................................................................................................................................144Εκτενής Περίληψη....................................................................................................................147

iii

CHAPTER 1: INTRODUCTION

CHAPTER 1

INTRODUCTION

1


1.1. Introduction

At present, the most important breast imaging technique is x-ray mammography.

Mammography must be capable to reveal not only subtle differences in the density and

composition of breast parenchymal tissue, but also the presence of minute calcifications

typically 100 μm in dimension. It is obvious that there is the need not only to maximize

the subject contrast, for the detection of soft-tissue lesions, but also to obtain a high degree

of resolution and low level of noise. In addition, due to the risks of ionizing radiation, the

dose in the breast should be kept ‘as low as reasonably achievable’ according to the

ALARA concept (ICRP 1991).

In trying to satisfy the above objectives and increase the sensitivity and specificity of

the mammographic procedure, a fact that would led to a more accurate diagnosis and

earlier breast cancer detection, research focuses on (a) the so-called computer-aided

diagnosis (CAD), which deals with the application of image processing, image analysis

and machine vision techniques on digitized mammographic images and (b) the

optimization of image quality and the minimization of dose in breast with the design and

refinement of dedicated mammographic equipment as well as the determination of

optimum standards for the operational parameters of a mammography unit. The research

in computer-aided diagnosis has a very remarkable progress to demonstrate. Nevertheless,

its success depends on the quality of the mammographic image obtained at the x-ray

image detector.

The x-ray image detector is one of the most important factors that affect the

efficiency of the mammographic technique. Xeroradiography was the first step in

producing mammographic images (Boag (1973)). It was introduced in the early 1970s and

employed an amorphous selenium plate as the image sensor. The development of the

screen-film mammography though offered superior performance in imaging low-contrast

structures with distinct boundaries. This limitation of xeroradiography was largely due to

the powder cloud development method used at the time. Screen-film mammography is still

the gold standard in the examination of the female breast. Despite this fact, its dynamic

range is limited (1:25) whereas masses and microcalcifications, important indicators of

cancer, are hardly visualized in very dense breasts.

Recent research has shown that digital mammography systems offer improved image

quality as compared to screen-film systems as well as increased quantum efficiency,

flexible image acquisition, processing and storage. Active matrix flat panel systems with

an electroded x-ray phosphor as detection material have proved to be superior to other

2


digital mammographic imaging modalities such as photostimulable phosphors and charge

coupled devices (CCDs) (Zhao et al 1997, Zhao and Rowlands 1995). In particular, direct

conversion digital flat panel systems, which directly convert x-rays to a charge cloud that

is electrically driven and stored in the pixels, provide improved quantum efficiency,

reduced blurring and high spatial resolution.

Among the most important components of direct detectors is the sensitive to the

radiation material (photoconductor). Amorphous selenium (a-Se) is one of the most

suitable materials mainly due to its ability to be coated over large areas with uniform

imaging characteristics and due to its high intrinsic spatial resolution. Nevertheless, this

material suffers from low x-ray absorption efficiency and x-ray sensitivity. Kasap and

Rowlands (2000) discussed the properties of an ideal x-ray photoconductor for a direct

conversion digital flat panel x-ray image detector. Materials like a-As2Se3, GaSe, GaAs,

Ge, CdTe, CdZnTe, Cd0.8Zn0.2Te, ZnTe, PbO, TlBr, PbI2 and HgI2 satisfy some of these

ideal characteristics and therefore are also potential candidates. Among them, the

polycrystalline materials CdTe, CdZnTe, Cd0.8Zn0.2Te, PbO, PbI2 and HgI2 as well as the

amorphous material a-As2Se3 are the most feasible candidates mainly due to the fact that

they can be grown in large areas. On the other hand, the crystalline materials GaSe, GaAs,

Ge, ZnTe and TlBr are difficult to be developed at such large areas with current

techniques and therefore are less suitable. Nevertheless, the current preparation procedures

are prone to improve.

To optimize the image quality and hence the diagnostic information acquired from

direct detectors, a careful selection of the photoconducting material must be made with the

simultaneous refinement of detector technology. This can be achieved with the

investigation of the physics that governs the signal formation processes in the

photoconductors mentioned since in this way important information relevant to the

production of the final image is acquired. Research has mainly focused on the lag and

ghosting phenomena (Bloomquist et al 2006, Bakueva et al 2006, Zhao et al 2005,

Zhao and Zhao 2005), the x-ray sensitivity and photogeneration (Steciw et al 2002,

Stone et al 2002, Street et al 2002, Kabir and Kasap 2002a, Kabir and Kasap 2002b,

Kasap 2000, Blevis et al 1999) as well as the charge carrier drifting, multiplication,

recombination and collection (Lui et al 2006, Cola et al 2006, Su et al 2005,

Kasap et al 2004, Kabir and Kasap 2004, Miyajima 2003, Hunt et al 2002,

Mainprize et al 2002, Miyajima et al 2002, Sato et al 2002, Kabir and Kasap 2002a,

3


Fourkal et al 2001, Lachaine and Fallone 2000a, Lachaine and Fallone 2000b,

Street et al 1999, Jahnke and Matz 1999).

1.2. Thesis

The quality of the mammographic image is directly related to its characteristics. The x-ray

induced primary electrons inside the photoconductor’s bulk comprise the primary signal

which propagates in the material and forms the final signal (image) at the detector’s

electrodes. Consequently, the characteristics of the mammographic image strongly depend

on the characteristics of the primary electrons. Experimentally is not feasible to study

exclusively the primary electrons. On the other hand, simulation studies in the materials

mentioned have not dealt with the characteristics of primary electrons such as their

number as well as their energy, angular and spatial distributions and furthermore with

their influence on the characteristics of the final image.

In this PhD thesis an investigation has been carried out concerning the primary

signal formation processes and the characteristics of primary electrons inside the

photoconducting materials mentioned. In addition, the influence of the characteristics of

primary electrons on the characteristics of the final signal together with the electric field

distribution and the electron interaction mechanisms particularly for the case of a-Se, one

of the most preferable photoconductors, have been studied at a first stage. The electric

field distribution and the electron interactions are two crucial parameters in the

development of a model that would simulate the final signal formation and hence study

the influence of the characteristics of the primary electrons on the characteristics of the

final image.

In particular, a Monte Carlo model that simulates the primary electron production

inside a-Se, a-As2Se3, GaSe, GaAs, Ge, CdTe, CdZnTe, Cd0.8Zn0.2Te, ZnTe, PbO, TlBr,

PbI2 and HgI2 has been developed. The model simulates the primary photon interactions

(photoelectric absorption, coherent and incoherent scattering), as well as the atomic

deexcitations (fluorescent photon production, Auger and Coster-Kronig electron

emission). The results, obtained for both monoenergetic and polyenergetic x-ray spectra in

the mammographic energy range, are grouped in four categories:

A. Energy distributions of: (i) fluorescent photons, (ii) primary and fluorescent

photons escaping forwards and backwards, (iii) primary electrons.

B. Azimuthal and polar angle distributions of primary electrons.

C. Spatial distributions of primary electrons.

4


D. Arithmetics of: (i) fluorescent photons, (ii) primary and fluorescent photons

escaping forwards and backwards, (iii) primary electrons.

In addition, a mathematical formulation has been developed for the drifting of

primary electrons of a-Se in vacuum under the influence of a capacitor’s electric field and

the resulting electron energy, angular and spatial distributions on the collecting electrode

have been studied. The formulation has been based on the Newton’s equations of motion

and the theorem for kinetic energy change.

Furthermore, the electric field distribution of Pang et al (1998) for a-Se detectors has

been adopted and reexamined to adjust it to the simulation model of primary electrons.

A code has been developed that calculates the distribution of the electric potential

anywhere in a-Se over the pixel and the pixel gap, using the analytical solution of Pang,

the boundary values of our case and appropriate numerical calculation methods.

Finally, the structure and the mathematical formulation of a model that would

simulate the electron interactions inside a-Se have been developed. They were based on

the model of Fourkal et al (2001) that has been reexamined and enriched with existing

theoretical considerations, developed mainly by Ashley (1988), and simulation

formalisms, developed mainly by Salvat et al (1985, 1987, 2003).The formulation has

included the electron free path length, the decision on the type of electron interaction, the

differential and total elastic scattering cross section and the differential and total inelastic

scattering cross sections with inner shells (K and L shells) as well as with outer shells.

Based on the results of primary electron production, a comparative study between

the various photoconductors is made concerning the number and the energy of fluorescent

and escaping photons as well as the number, the energy and the angular and spatial

distributions of primary electrons. Studying the primary electron production for the

monoenergetic case, insights are gained into the related physics that lead to the

investigation of the primary electron characteristics as well as the factors which affect

them. The polyenergetic case provides information about the dependence of these

characteristics on the incident mammographic spectrum. Moreover, the results obtained

for a-Se primary electrons that drift in vacuum under the influence of a capacitor’s electric

field and are being collected from the top electrode, although they pertain to an unrealistic

case, yet give at a first approximation the influence of the characteristics of the primary

signal on the characteristics of the final signal. Finally, the formulations for the electric

field distribution and the electron interactions inside a-Se, can form the basis of

developing a simulation model for the signal propagation inside the photoconductor’s

5


bulk, a fact that would help to derive conclusive remarks on the correlation of primary and

final signal characteristics and hence optimize the performance of direct detectors as well

as select the most suitable materials for this kind of applications.

1.3. Thesis layout

The layout of this PhD thesis has been built as follows:

The first two chapters deal with mammographic imaging issues. Chapter 2 describes

briefly the mammographic imaging technique as well as the mammographic equipment

whereas chapter 3 discusses and compares screen-film systems with digital

mammographic detectors. Chapter 4 analyses the properties of some of the most suitable

photoconductors for active matrix flat panel direct detectors such as a-Se, CdTe, CdZnTe,

PbO, HgI2 and PbI2. The theoretical background of the physics related to the image

formation processes is the subject of chapter 5. A detailed analysis of the physics of

x-ray-matter interactions, atomic deexcitation mechanisms and electron interactions is

made with a further discussion on charge carrier transport and recombination mechanisms

inside a-Se. Chapter 6 discusses the mathematical foundation of Monte Carlo calculations

and describes the basic Monte Carlo simulation methods. Chapters 7 and 8 deal with the

x-ray induced primary electrons inside the selected photoconductors. The modeling of

primary electron production is the subject of chapter 7 whereas the obtained results are

presented and discussed in chapter 8. Chapter 9 presents the mathematical formulation for

the drifting of primary electrons of a-Se in vacuum under the influence of a capacitor’s

electric field and discusses the effect of this drifting on their characteristics. The electric

field distribution and electron interactions inside a-Se are the subjects of chapters 10

and 11. The calculation method with the relevant analytical solution of Pang et al (1998)

for the electric field distribution inside a-Se detectors as well as a derived electric potential

distribution are presented in chapter 10. Chapter 11 gives the structure and the

mathematical formulation of the simulation model for electron interactions inside a-Se.

Finally, chapter 12 discusses the conclusions drawn in this PhD thesis as well as the

research work that will be conducted in the near future.

1.4. Publications

The research conducted during this PhD thesis resulted in publications in international

journals and international conference proceedings.

6


Publications in peer reviewed international journals:

1. Sakellaris T, Spyrou G, Tzanakos G and Panayiotakis G 2005 Monte Carlo simulation

of primary electron production inside an a-selenium detector for x-ray mammography:

physics Phys. Med. Biol 50 3717-38.

2. Sakellaris T, Spyrou G, Tzanakos G and Panayiotakis G 2007 Energy, angular and

spatial distributions of primary electrons inside photoconducting materials for digital

mammography: Monte Carlo simulation studies Phys. Med. Biol 52 6439-60.

3. Sakellaris T, Spyrou G, Tzanakos G and Panayiotakis G 2008 Photon and primary

electron arithmetics in photoconductors for digital mammography: Monte Carlo

simulation studies Nucl. Instrum. Methods A (accepted)

Publications in international conference proceedings:

1. Sakellaris T., Spyrou G., Tzanakos G. and Panayiotakis G. “Digital Mammography

using a-Se: Monte Carlo Generated Energy and Spatial Distributions of Primary

Electrons”, X Mediterranean Conference on Medical and Biological Engineering and

Computing, August 2004, Ischia, Italy.

2. Sakellaris T., Spyrou G., Tzanakos G. and Panayiotakis G. “Distributions of x-ray

Generated Primary Electrons in a-Se: Monte Carlo Simulation Studies”, 1st

International Conference "From Scientific Computing to Computational Engineering",

September 2004, Athens, Greece.

1.5. Financial support

This PhD research work was supported by a grant from the State Scholarship Foundation

of Greece (ΙΚΥ).

7

CHAPTER 2: MAMMOGRAPHY

CHAPTER 2

MAMMOGRAPHY

8


2.1. Introduction

Mammography is the examination of the female breast by the use of x-rays. The small

x-ray tissue attenuation differences in the breast require the use of equipment specifically

designed to demonstrate low contrast and fine detail at the same time. Due to the risks of

ionizing radiation, techniques that minimize dose and optimize image quality are very

important. Clinically, the determination of optimum standards for the operational

parameters of a mammographic unit is crucial. In this chapter, the mammographic

technique and the basic mammographic equipment are briefly discussed in conjunction

with mammographic image quality issues.

2.2. Mammographic Equipment

The mammographic unit consists of two basic components mounted on opposite sides of a

mechanical assembly: an x-ray tube and an image receptor. To accommodate patients of

different height and due to the fact that the breast must be imaged from different aspects,

the assembly can be adjusted vertically and can rotate about a horizontal axis. The

system’s geometry is arranged as shown in figure 2.1. The radiation leaves the x-ray tube

and passes through a metallic spectral-shaping filter, a beam-defining aperture and a plate

that compresses the breast. The rays coming out from the breast can either be absorbed by

an antiscatter grid or impinge on the image receptor. A fraction of x-rays passes through

the receptor without interaction and is incident on a sensor used to activate the automatic

exposure control mechanism of the unit (Yaffe 1995).

9

X-ray tube

Compression device

Image receptor


Figure 2.2. A typical mammographic x-ray tube.

2.2.1. X-ray tube

The x-rays used in mammography arise from bombardment of a metal target (anode) by

electrons in a hot-cathode vacuum tube. The x-rays are emitted from the target over a

spectrum of energies ranging up to the peak kilovoltage (kVp) applied to the x-ray tube

(typically 30 kVp). A rotating anode design is used for modern mammographic x-ray

tubes. The most common targets are those made of molybdenum (Mo). Nevertheless,

targets made of tungsten (W), rhodium (Rh) or alloys combining these elements are being

used as well. The anode has a beveled edge, which is at a steep angle to the direction of

the electron beam. The exit window accepts x-rays that are approximately at right angles

to the electron beam so that the x-ray source as viewed from the receptor appears to be

approximately square even though the incident electron beam is slit-shaped. The

resolution and optimal image quality required in mammography demands the use of very

small focal spots for contact and magnification imaging. Typical focal spot sizes range

from 0.3 to 0.4 mm for contact imaging and from 0.1 to 0.15 mm for magnification

imaging. Most mammography tubes use beryllium (Be) windows between the evacuated

tube and the outside world because glass or other metals would provide excessive

attenuation of the useful energies for mammography. Since the high and low energies in

the spectrum are suboptimal in terms of imaging the breast, added filtration (usually of the

10

Figure 2.1. A typical mammographic unit.


same element as the target) selectively attenuates and optimizes the beam spectrum.

Collimation of the x-ray beam is accomplished usually by diaphragms. Diaphragm

collimators are metal apertures with predetermined field sizes matched to the image

receptor’s sizes (i.e. 18 x 24 cm2 or 24 x 30 cm2). Figure 2.2. presents a typical

mammographic x-ray tube.

2.2.2. Compression device

The compression to the breast is achieved with a compression paddle, a flat plate attached

to a mechanical compression device. It is essential that the compression plate allows the

breast to be compressed parallel to the image receptor and that the edge of the plate at the

chest wall be straight and aligned with both the focal spot and image receptor to maximize

the amount of breast tissue being included in the image. Compression causes the different

tissues to be spread out, minimizing superposition from different planes and thereby

improving conspicuity of structures. The use of compression decreases geometric blurring,

the dose to the breast and the ratio of scattered to directly transmitted radiation that

reaches the image receptor.

2.2.3. Antiscatter grid

Scattered radiation comprises a considerable fraction of the radiation incident on the

image receptor. It degrades the subject contrast according to the following rule

(Yaffe 1995): Cs=Co/(1+SPR), where Cs is the subject contrast, Co is the contrast in the

absence of scattered radiation and SPR is the Scatter-to-Primary x-ray Ratio at the location

of interest in the image. The fraction of scattered radiation in the image can be reduced by

the use of antiscatter grids or air gaps. Scatter rejection is best accomplished with an

antiscatter grid for contact film/screen breast imaging. Antiscatter grids are composed of

linear lead (Pb) septa separated by a rigid interspace material (usually paper). Generally,

the grid septa are not strictly parallel but are focused toward the x-ray source. Due to the

fact that the primary x-rays all travel along direct lines from the x-ray source to the image

receptor while the scatter diverges from points within the breast, the grid presents a

smaller acceptance aperture to scattered radiation than to primary and therefore

discriminates against scattered radiation. Grids are characterized by their grid ratio (ratio

of the path length through the interspace material to the interseptal width), which typically

ranges from 3.5:1 to 5:1. When a grid is used, the SPR is reduced approximately by a

factor of about 5, leading in the most cases to a significant improvement in image contrast.

11


Nevertheless, the grid causes the overall radiation fluence to decrease. To compensate for

losses and therefore to obtain a mammogram of proper optical density, the entrance

Figure 2.3. The geometrical characteristics of an antiscatter grid: h the height of lead

strips, d the thickness of lead strips, D the thickness of paper, 1/(D+d) the strip density

and h/D the grid ratio.

exposure to the patient is increased by a factor typically between 2 and 3 known as the

Bucky factor. Figure 2.3. presents the geometrical characteristics of an antiscatter grid.

2.2.4. Image receptors

The image receptor forms the image by the absorption of energy from the x-ray beam.

Image receptors must provide adequate spatial resolution, radiographic speed and

image contrast. There is a variety of techniques used to visualize the distribution of energy

being absorbed inside the receptor. The detectors are divided into two categories:

(a) analog detectors (for example a high resolution fluorescent screen in conjunction with

a radiographic film) that “reconstruct” the distribution in a continuous manner in the

intensity scale and (b) digital detectors (for example an a-Se detector with an active matrix

flat panel system) that sample the distribution in space and in intensity scale. The detectors

used in mammography are discussed in detail in the next chapter.

12

CHAPTER 2: MAMMOGRAPHY 13

CHAPTER 3: MAMMOGRAPHIC DETECTORS

CHAPTER 3

MAMMOGRAPHIC DETECTORS

14


3.1. Introduction

As stated in the previous chapter the image receptor (detector) forms the image by the

absorption of energy from the x-ray beam. The mammographic detectors are divided into

two categories: (a) analog detectors and (b) digital detectors. In this chapter a detailed

comparative description of the structure and performance of both types of detectors is

made.

3.2. Phosphors

Most mammographic image receptors employ a phosphor at an initial stage to convert

x-rays into visible light. Phosphor screens are typically produced by combining 5–10 μm

diameter phosphor particles with a transparent plastic binder. The use of phosphor

materials with a relatively high atomic number causes the photoelectric effect to be the

dominant type of x-ray interaction. The energy of an x-ray is much larger than the

bandgap of the phosphor crystal and, therefore, in being stopped, a single interacting x-ray

has the potential to cause the excitation of many electrons in the bulk and thereby the

production of many light quanta. After their production, the light quanta must successfully

escape the phosphor and be effectively coupled to the next stage in image formation. It is

desirable to ensure that the created light quanta escape the phosphor efficiently and as near

as possible to their point of formation. Since the probability of x-ray interaction is

exponential, the number of interacting quanta and the amount of light created will be

proportionally greater near the x-ray entrance surface (Yaffe and Rowlands 1997).

3.3. Film-Screen systems (Analog detectors)

The radiographic film is the most widely used detector in diagnostic radiology. The most

important component in a radiographic film is the sensitive to the radiation layer called

‘emulsion’. The emulsion comprises of a gelatin in which AgBr or AgI grains are embed.

Films have either single or double emulsions.

In routine mammography, nowadays films combined with a phosphor screen made

of CaWO4 or Gd2O2S:Tb are used almost exclusively as image receptors. The application

of non-screen x-ray film is either no longer recommended or explicitly forbidden because

of the high radiation exposure. Often the sensitivity of the image receptor is characterized

by the so-called ‘system dose’, which is usually defined as the air kerma at the location of

the image receptor needed to obtain the receptor-specific exposure. The system dose of a

15


Figure 3.1. Cross section of a simplified film-screen detector.

modern film–screen system is about 1–3% of that of the non-screen film (Säbel and

Aichinger 1996). In mammography, films are always single sided and used with a back

screen only. This is to avoid any possibility of ‘parallax unsharpness’ which may arise

from the two images in double sided film (Law 2006). In figure 3.1 the cross section of a

simplified film-screen detector is shown.

3.4. Digital Detectors

In order to generate a digital x-ray image, the intensity of the incident x-ray beam must be

sampled in both the intensity and spatial domains. In the intensity domain, the magnitude

of the x-ray intensity is converted into a proportional electronic signal; this signal is then

digitized so that it can be sent to a computer where the final image will be processed. In

the spatial domain, the variation in the intensity signal over the area of the object

represents the image information. Therefore, it is necessary to coordinate the digitized

intensity signal with its position within the active imaging area of the detector. Any digital

radiography solution therefore consists of two parts: the conversion of incident x-ray

photons into an electrical signal, and the measurement of the spatial variation in this

signal. The digital detectors used in mammography are divided into three categories:

A. Phosphor-charge coupled devices (CCD) systems.

B. Photostimulable phosphors.

C. Active Matrix Flat Panel Imagers (AMFPI).

3.4.1. Phosphor-CCD systems

3.4.1.1. CCD devices

Charge coupled devices are particularly well suited to digital radiography because of their

high spatial resolution capability, wide dynamic range and high degree of linearity with

16

Incident X-rays

Film

Phosphor screen Visible light


Figure 3.2. The structure of a CCD array, illustrating motion of stored charge in one

direction as the potential wells are adjusted under control of the gate electrode voltages

(Yaffe and Rowlands 1997).

incident signal. They can be made sensitive to light or to direct electronic input. A CCD

(figure 3.2) is an integrated circuit formed by depositing a series of electrodes, called

‘gates’, on a semiconductor substrate to form an array of metal-oxide-semiconductor

(MOS) capacitors. By applying voltages to the gates, the material below is depleted to

form charge storage ‘wells’. These store charge injected into the CCD or generated within

the semiconductor by the photoelectric absorption of optical quanta. If the voltages over

adjacent gates are varied appropriately, the charge can be transferred from well to well

under the gates, much in the way that boats will move through a set of locks as the

potentials (water heights) are adjusted (Yaffe and Rowlands 1997).

3.4.1.2. Optical coupling of a phosphor to a CCD

A phosphor can be coupled to a CCD either by a lens/mirror system (figure 3.3(a)) or fibre

optics (figure 3.3(b)).

In a lens/mirror system, a fraction of the emitted light is reflected on a mirror or is

driven directly to a lens that guides the light onto the CCD. Because the size of available

CCDs is limited from manufacturing considerations to a maximum dimension of only

2–5 cm, it is often necessary to use a demagnifying lens.

In the case of fibre optics, which can be in the form of fibre optic bundles, optical

fibres of constant diameter are fused to form a light guide. The fibres form an orderly

array so that there is a one-to-one correspondence between the elements of the optical

17


Figure 3.3. The two ways of coupling a phosphor to a CCD: (a) optical coupling and

(b) fibre optic coupling (Yaffe and Rowlands 1997).

image at the exit of the phosphor and at the entrance to the CCD. To accomplish the

required demagnification, the fibre optic bundle can be tapered by drawing it under heat

(Yaffe and Rowlands 1997). Systems of both designs are used in cameras with a small

field of view for digital mammography. In such applications, much lower

demagnification, typically two times, is used, resulting in acceptable coupling efficiency.

By abutting several camera systems to form a larger matrix, a full-field digital breast

imaging system can be constructed similar to that presented in figure 3.4.

Figure 3.4. A full-field digital breast imaging system composed of a matrix of phosphors

coupled to CCDs by fibre optics.

18

(a) (b)

Phosphor

CCDDemagnifying

fibre optic taper


3.4.1.3. Slot scanned digital mammography

In order to overcome the size and cost limitations of available high-resolution

photodetectors in producing a large imaging field (like the one presented in figure 3.4) slot

scanned digital mammographic systems have been developed (figure 3.5).

Figure 3.5. A phosphor–fibre optic–CCD detector assembly for slot-scanned digital

mammography (Pisano and Yaffe 2005).

In these systems the detector has a long, narrow, rectangular shape, with dimensions of

approximately 1 x 24 cm2 and the x-ray beam is collimated into a narrow slot to match this

format. Acquisition takes place in time delay integration (TDI) mode in which the x-ray

beam is activated continuously during the image scan and charge collected in pixels of the

CCDs is shifted down CCD columns at a rate equal to but in the opposite direction as the

motion of the x-ray beam and detector assembly across the breast. The collected charge

packets remain essentially stationary with respect to a given projection path of the x-rays

through the breast and the charge is integrated in the CCD column to form the resultant

signal. When the charge packet has reached the final element of the CCD, it is read out on

a transfer register and digitized (Pisano and Yaffe 2005, Yaffe and Rowlands 1997).

.

3.4.2. Photostimulable phosphors (Computed radiography systems)

Photostimulable phosphors are commonly in the barium fluorohalide family, typically

BaFBr:Eu+2, where the atomic energy levels of the europium activator determine the

characteristics of light emission. X-ray absorption mechanisms are identical to those of

conventional phosphors. They differ in that the useful optical signal is not derived from

the light that is emitted in prompt response to the incident radiation, but rather from

subsequent emission when electrons and holes are released from traps in the material. The

initial x-ray interaction with the phosphor crystal causes electrons to be excited. Some of

19


Figure 3.6. A schematic of a photostimulable phosphor digital radiography system

(Yaffe and Rowlands 1997).

these produce light in the phosphor in the normal manner, but the phosphor is intentionally

designed to contain traps which store the charges. By stimulating the crystal by irradiation

with red light, electrons are released from the traps and raised to the conduction band of

the crystal, subsequently triggering the emission of shorter-wavelength (blue) light. This

process is called photostimulated luminescence. In the digital radiography application

(figure 3.6), the imaging plate is positioned in a light-tight cassette or enclosure, exposed

and then read by raster scanning the plate with a laser to release the luminescence. The

emitted light is collected and detected with a photomultiplier tube whose output signal is

digitized to form the image (Yaffe and Rowlands 1997). Photostimulable phosphors are

widely used in digital mammography because:

When placed in a cassette, they can be used with conventional x-ray machines.

Large-area plates are conveniently produced, and because of this format, images

can be acquired quickly.

The plates are reusable, have linear response over a wide range of x-ray

intensities, and are erased simply by exposure to a uniform stimulating light source to

release any residual traps.

Nevertheless, there are certain disadvantages as well:

Due to the fact that the traps are located throughout the depth of the phosphor

material, the laser beam providing the stimulating light must penetrate into the

phosphor. Scattering of the light within the phosphor causes release of traps over a

20


greater area of the image than the size of the incident laser beam. This results in loss of

spatial resolution.

The readout stage is mechanically complex, and efficient collection of the emitted

light requires great attention to design (Yaffe and Rowlands 1997).

3.4.3. Active matrix flat panel imagers (AMFPI)

The active matrix flat panel technology is the most promising digital radiographic

technique (Kasap and Rowlands 2000, Yaffe and Rowlands 1997). It is based on large

glass substrates on which imaging pixels are deposited. The term ‘active matrix’ refers to

the fact that the pixels are arranged in a regular two-dimensional grid, with each pixel

containing an amorphous silicon (a-Si:H) based switch that is usually a thin film transistor

(TFT). The pixel switch is connected to some form of pixel storage capacitor that serves to

hold an imaging charge induced by the incident radiation. The typical dimension of a flat

panel system for digital mammography is 18 x 24 cm2 (Zhao and Rowlands 1995). There

are two approaches for this kind of systems: the indirect and direct detection.

3.4.3.1. Indirect detection

In figure 3.7(a) and 3.7(b) the cross section and a microphotograph of a pixel of an

indirect AMFPI is presented.

Figure 3.7. (a) A cross section and (b) a microphotograph of a pixel for an indirect

AMFPI (Antonuk et al 2000).

In the indirect detection approach, a conventional x-ray absorbing phosphor, such as

Gd2O2S, is placed or thallium-doped caesium iodide (CsI:Tl) is grown onto the active

21

(a) (b)


matrix array. The detector pixels are configured as photodiodes (made of a-Si:H) which

convert the optical signal from the phosphor to charge and store that charge on the pixel

capacitance. The advantage of utilizing CsI as the x-ray absorber is that it can be grown in

columnar crystals which act as fibre optics. When coupled to the photodiode pixels, there

is little lateral spread of light and, therefore, high spatial resolution can be maintained. In

addition, unlike conventional phosphors in which diffusion of light and loss of resolution

become worse when the thickness is increased, CsI phosphors can be made thick enough

to ensure high x-ray absorption while maintaining high spatial resolution

(Yaffe and Rowlands 1997). The signal readout in the active matrix is the same for both

the indirect and direct method and hence it is described in the next section.

3.4.3.2. Direct detection

In a direct detector for digital mammography, a high atomic number photoconductor (for

example a-Se or PbI2) is coated onto the active matrix area to form a photoconducting

layer that directly converts the incident x-rays into charge carriers that drift towards the

collecting electrodes under the influence of an applied electric field.

The direct detection systems have advantages compared to the indirect systems. As

discussed earlier, x-rays absorbed in the screen of an indirect system release light which

must escape to the surface to create an image while lateral spread of light is determined by

diffusion. Thus, the blur diameter is comparable to the screen thickness. This blurring

causes a loss of high-frequency image information which is fundamental and largely

irreversible. Although the loss can be alleviated when using CsI, the separation between

fibres is created by cracking and as a result the channeling of light is not perfect. In the

case of direct detection, since the produced charges are electrically driven towards the

electrodes, their lateral spread, and hence the image blurring, is not significant

(Yaffe and Rowlands 1997). Furthermore, the absorption efficiency of a direct detector

can be maximized with the suitable choice of the photoconductor material, operating bias,

and the thickness of the photoconductive layer (Kasap 2000). Finally, the direct systems

are easier and cheaper to manufacture due to their simpler structure (Saunders et al 2004,

Samei and Flynn 2003). The major disadvantages of direct detectors are the need for

applying a high voltage to maintain the electric field and the presence of dark current

(Pisano and Yaffe 2005).

22


A microphotograph and a simplified physical structure of a single pixel with TFT as

well as a simplified schematic diagram of the cross sectional structure of two pixels of a

direct conversion x-ray detector are shown in figure 3.8.

An Indium Tin Oxide (ITO) electrode (labeled A) is uniformly deposited on the

photoconducting layer usually with thermal evaporation. This electrode is called the ‘top

electrode’. The top electrode is positively biased with a high voltage to create an electric

field in the photoconductor’s bulk that has a typical value of 10 V/μm. Amorphous

selenium (a-Se) is the most highly developed photoconductor for direct applications due to

its amorphous state, that makes possible the maintenance of uniform imaging

characteristics to almost atomic scale (there are no grain boundaries) over large areas, and

due to its high intrinsic resolution that can exceed 500 lp/mm (Que and Rowlands 1995a).

Typical values for the photoconductor thickness (when using a-Se) ranges from 200 to

500 μm (Pang et al 1998). Similar to the top electrode, the photoconducting layer is

thermally evaporated onto the active matrix. As mentioned earlier, when x-rays are

absorbed in the photoconductor’s bulk, electron-hole pairs are created which under the

influence of the electric field separate. Thus, the electrons drift towards the top electrode

while the holes towards the active matrix where they are collected and stored.

The active matrix consists of M x N pixels (for example 3600 x 4800 pixels,

Zhao and Rowlands 1995). Each pixel has three basic elements: the TFT switch, the pixel

electrode and the storage capacitor. The standard configuration is the one presented in

figure 3.8(b) where the TFT, the pixel electrode and the capacitor are in the same level.

The active matrix is characterized by the pixel width (a), the pixel collection width (acoll)

and the pixel pitch (d) (figure 3.8(b)). The typical pixel pitch for mammography is 50 μm

(Zhao and Rowlands 1995). The ratio Fcoll= a2coll/a2 is defined as the collection fill factor

whereas the ratio Fgeom=a2/d2 as the geometric fill factor (Antonuk et al 2000). In order to

increase the Fcoll some systems incorporate the storage capacitor and the TFT underneath

the pixel electrode. This pixel structure is known as the ‘mushroom structure’

(Pang et al 1998). The pixel voltage Vp increases as a function of the x-ray exposure.

Normally this voltage does not exceed 10 V. Nevertheless, under suspended scans or

accidental overexposures it can reach up to values similar to the high voltage applied on

the top electrode a fact that damages the detector. To protect the detector from high

voltages, an insulating layer is placed either between the top electrode and the

photoconductor or between the photoconductor and the active matrix. In this way the

trapped charges in the interface between the insulating layer and the photoconductor

23


Figure 3.8. Direct conversion x-ray detectors: (a) a microphotograph (Antonuk et al

2000) and (b) a simplified physical structure of a single pixel with TFT (Kasap and

Rowlands 2000). (c) A simplified schematic diagram of the cross sectional structure of

two pixels (Kasap and Rowlands 2002).

decrease the electric field and hence the VP saturates (Zhao and Law 1998). In addition,

this insulating layer prevents charges from either the top electrode or the active matrix to

be injected into the photoconductor’s bulk and reduces aliasing (Zhao and

24

acoll

Pixel width (a)

Pixel pitch (d)

(a) (b)

(c)


Rowlands 1997). The thickness of the insulating layer is (for the case of a-Se) one-tenth of

that for the photoconductor.

As mentioned earlier, the TFTs act as switches on each individual pixel and control

the image charge so that one line of pixels is activated electronically at a time. Normally

all TFTs are turned off permitting the charges to accumulate on the pixels electrodes. The

readout is achieved by external electronics and software control of the state of TFTs

(Kasap and Rowlands 2002). Each TFT has three electrical connections as shown in figure

3.8(b): the gate (G) for the control of the ‘on’ or ‘off’ state of the TFT, the drain (D) that is

connected to the pixel electrode and a pixel storage capacitor, and the source (S) that is

connected to a common data line. When gate line i is activated, all TFTs in that row are

turned on and N data lines from j = 1 to N then read the charges on the pixel electrodes in

row i. The parallel data are multiplexed into serial data, digitized, and then fed into a

computer for imaging. The scanning control then activates the next row (i + 1) and all the

pixel charges in this row are then read and multiplexed, and so on until the whole matrix

has been read from the first to the last row (M-th row). It is apparent that the charge

distribution residing on the panel's pixels is simply read out by self-scanning the arrays

row-by-row and multiplexing the parallel columns to a serial digital signal. This signal is

then transmitted to a computer system (Kasap and Rowlands 2000). Table 1 summarizes

the required specifications for flat panel detectors for digital mammography.

Table 3.1. Parameters for digital x-ray imaging systems (Zhao and Rowlands 1995).

3.5. Digital versus Film-Screen Mammography

Film–screen mammography is still the gold standard for the detection and diagnosis of

breast cancer. Film–screen combinations provide excellent detail resolution (image

sharpness), which is very crucial for imaging microcalcifications and very small

25

Detector parameter Value

Detector size (cm2) 18 x 24Pixel pitch (μm) 50Number of pixels 3600 x 4800Readout time (s) < 5X-ray spectrum (kVp) 30Mean exposure (mR) 12Exposure range (mR) 0.6-240


abnormalities that may indicate early breast cancer. Nevertheless, film-screen systems

have certain limitations (Säbel and Aichinger 1996, Yaffe and Rowlands 1997):

Narrow dynamic range (1:25), which must be balanced against the need for wide

latitude (1:100). The slope of the characteristic curve of the radiographic film

determines the contrast properties and the attenuation difference between a lesion and

the surrounding tissue which can be seen in the image. Masses and microcalcifications

are therefore hardly visualized in very dense breasts by film–screen combinations.

Noise associated with film granularity.

Inefficient use of the incident radiation.

By contrast with the film–screen technique, in digital mammography with the help

of the windowing technique the detectability of subtle details is limited only by noise.

Digital mammography thus has the potential for improving the display of poorly

contrasted details. Comparative studies between digital and screen-film mammography

have shown that digital mammography provides similar image quality

(Haus and Yaffe 2000, Lewin et al 2001) and sometimes better (Obenauer et al 2002)

compared to screen-film mammography in terms of detecting breast cancer. Digital

mammography has additional advantages over conventional mammography

(Berns et al 2002, Huda et al 2003):

Wider dynamic range.

The magnification, orientation, brightness and contrast of the mammographic

image can be altered after the examination has completed.

Improved contrast between dense and non-dense breast tissue.

Faster image acquisition (less than a minute).

Shorter examination time (approximately half of the time in conventional

mammography).

Easier image storage.

Transmission of images for remote consultation with other physicians.

Nevertheless, digital mammography has also disadvantages as compared to screen-film

mammography:

It presents lower image sharpness.

It is expensive.

A method must be developed to compare digital mammographic images with

existing screen-film images on computer monitors.

26

CHAPTER 4: X-RAY PHOTOCONDUCTORS

CHAPTER 4

X-RAY PHOTOCONDUCTORS

27


4.1. Introduction

The performance of a direct digital mammographic detector strongly depends on the

properties of the photoconducting material used to convert the incident x-rays into charge

carriers. In this chapter the properties of suitable x-ray photoconductors are discussed and

compared with the ideal case.

4.2. The ideal x-ray photoconductor

Ideally the photoconducting layer should posses the following material properties (Kasap

and Rowlands 2000):

i. The photoconductor should have as high an intrinsic X-ray sensitivity as possible,

that is, it must be able to generate as many collectable (free) electron-hole pairs as

possible per unit of incident radiation. This means that the amount of radiation

energy required to generate a single free electron-hole pair ( ) must be as low as

possible.

ii. Nearly all the incident X-ray radiation should be absorbed within a practical

photoconductor thickness to avoid unnecessary exposure of the patient.

iii. There should be no dark current. This means the contacts to the photoconductor

should be non-injecting and the rate of thermal generation of carriers from various

defects or states in the bandgap should be negligibly small.

iv. There should be no bulk recombination of electrons and holes as they drift to the

collection electrodes.

v. There should be no deep trapping of electron-hole pairs which means that, for both

electrons and holes, the mean free distance (the average distance a carrier drifts

before it is trapped and unavailable for conduction) μτE>>L, where μ is the drift

mobility, τ is the deep trapping time (lifetime), E is the electric field and L is the

layer thickness.

vi. The longest carrier transit time, which depends on the smallest drift mobility, must

be shorter than the image readout time (pixel access time).

vii. The above should not change or deteriorate with time and as a consequence of

repeated exposure to x-rays. That is, x-ray fatigue and x-ray damage should be

negligible.

28


viii. The photoconductor should be easily coated onto the active matrix panel, for

example, by conventional vacuum techniques. Special processes are generally more

expensive.

ix. The photoconductor should have uniform characteristics over its entire area.

For the time being, the large area coating requirements in mammography (typically

over 30 × 30 cm2 or more), makes amorphous (a-) and polycrystalline (poly-)

photoconductors to be more suitable for digital detectors as compared to crystalline

materials which are difficult to grow in such large areas with current techniques. a-Se is

one of the most highly developed photoconductor due to its commercial use as an

electrophotographic photoreceptor. It can be easily coated as thick films (e.g. 100-500 μm)

onto suitable substrates by conventional vacuum deposition techniques and without the

need to raise the substrate temperature beyond 60-70 oC. In addition, its amorphous state

maintains uniform characteristics to very fine scales over large areas (Kasap and

Rowlands 2000, 2002a, 2002b). Nevertheless, due to its high compared to other

materials there has been an active research to find x-ray photoconductors that could

replace a-Se in flat panel image detectors (Kasap and Rowlands 2000, 2002a , 2002b).

4.3. Crystalline, amorphous and polycrystalline solids

A perfect elemental crystal consists of a regular spatial arrangement of atoms, with

precisely defined distances (the interatomic spacing) separating adjacent atoms. Every

atom has a strict number of bonds to its immediate neighbors (the coordination) with a

well defined bond length and the bonds of each atom are also arranged at identical angular

intervals (bond angle). This perfect ordering maintains a long range order and hence a

periodic structure (Kabir 2005). A hypothetical two-dimensional crystal structure is shown

in figure 4.1(a).

An amorphous solid exhibits no crystalline structure or long range order and it only

possesses short range orders because the atoms of an amorphous solid must satisfy their

individual valence bonding requirements, which leads to a little deviation in the bonding

angle and length. Thus, the bonding geometry around each atom is not necessarily

identical to that of other atoms, which leads to the loss of long-range order as illustrated in

figure 4.1(b). As a consequence of the lack of long-range order, amorphous materials do

not possess such crystalline imperfections as grain boundaries and dislocations, which is a

distinct advantage in certain engineering applications (Kabir 2005).

29


Figure 4.1. Two dimensional representation of the structure of (a) a crystalline solid

and (b) an amorphous solid.

A polycrystalline material is not a single crystal as a whole, but it is composed of

many small crystals randomly oriented in different directions. The small crystals in

polycrystalline solids are called grains. These grains have irregular shapes and orientations

and are separated by the so-called grain boundaries (figure 4.2). At a grain boundary,

atoms obviously cannot follow their normal bonding tendency and there exist voids,

stretched and broken bonds, as well as misplaced atoms which cannot follow the

crystalline pattern on either side of the boundary. In many polycrystalline materials,

impurities tend to congregate in the grain boundary region. The main drawbacks of

polycrystalline materials are the adverse effects of the grain boundaries which limit charge

transport, the nonuniform response of the sensor (pixel to pixel sensitivity variation) due

to large grain sizes, which reduces the dynamic range of the imagers, and the image lag

(Kabir 2005).

4.4. X-ray photoconductors

Materials like a-Se, a-As2Se3, GaSe, GaAs, Ge, CdTe, CdZnTe, Cd0.8Zn0.2Te, ZnTe, PbO,

TlBr, PbI2 and HgI2 satisfy some of the ideal characteristics mentioned earlier and hence

are potential candidates as photoconductors in direct detectors for digital mammography.

The most important ones are a-Se, CdTe, CdZnTe, PbO, PbI2 and HgI2 due their

amorphous and polycrystalline structure and due to certain properties that will be

discussed in detail. The information given is mainly obtained from Kabir (2005) and

Nesdoly (1999).

30

(a) (b)


Figure 4.2. (a) The grain structure of polycrystalline solids. (b) The grain boundaries

have impurity atoms, voids, misplaced atoms, and broken and strained bonds

(Kasap 2002).

4.4.1. Amorphous Selenium (a-Se)

Selenium is a member of the group VI column of the periodic table. The family name of

the elements of this group is chalcogens. The atomic number (Z) of selenium is 34, and it

has six valence electrons. Its electronic structure is [Ar]3d104s2p4. The density of a-Se is

4.3 g/cm3, relative permittivity εr = 6.7, and energy gap Eg = 2.22 eV.

Amorphous selenium can be quickly and easily deposited as a uniform thick film

(for example 100-1000 μm) over large areas (for example 40 cm × 40 cm or larger) by

conventional vacuum deposition techniques and without the need to raise the substrate

temperature beyond 60-70 °C. Nevertheless, pure a-Se is thermally unstable and

crystallizes over time. Alloying pure a-Se with As (0.2 – 0.5% As) greatly improves the

stability of the composite film and helps to prevent crystallization. However, it is found

that arsenic addition has adverse effect on the hole lifetime because the arsenic introduces

deep hole traps. If the alloy is doped with 10 – 20 parts per million (ppm) of a halogen

(such as Cl), the hole lifetime is restored to its initial value. Thus, a-Se film that has been

alloyed with 0.2 – 0.5% As (nominal 0.3% As) and doped with 10 – 20 ppm Cl is called

stabilized a-Se. Stabilized a-Se is currently the preferred photoconductor for clinical x-ray

image sensors. Crystalline Se is unsuitable as an x-ray photoconductor because it has a

much lower dark resistivity and hence orders of magnitude larger dark current than a-Se.

Stabilized a-Se has excellent transport properties, with typical hole and electron

ranges (μτ products) being 30 x 10-6 cm2/V and 5 x 10-6 cm2/V respectively. At typical

operating fields (>10 V/μm) the hole mean free path length is 30 mm whereas that for

electrons 5 mm. Since as mentioned in the previous chapter most a-Se detectors are

200-500 μm thick, these large mean free paths ensure that no free charges will be lost due

31


to trapping. The dark resistivity of a-Se is ~ 1014 Ω-cm. The dark current in a-Se detectors

is less than the acceptable level (1 nA/cm2) for an electric field as high as 20 V/μm. The

image lag in a-Se detectors is under 2% after 33 ms and less than 1% after 0.5 s in the

fluoroscopic mode of operation (Choquette et al 2001). Therefore, image lag in a-Se

detectors is considered as negligible. The pixel to pixel sensitivity variation is also

negligible in a-Se detectors.

The charge transport properties of a-Se as compared to a-As2Se3 are better for both

electrons and holes. In a-As2Se3 electrons are trapped and holes have much smaller

mobility. In addition, the dark current of a-Se is much smaller than a-As2Se3 (Kasap and

Rowlands 2000).

Where a-Se suffers in comparison to other materials is in two areas: x-ray absorption

and x-ray sensitivity ( ). Since the Z of Se is 34, a-Se is a rather poor absorber of x-rays

and a thicker detector must be used to absorb the same amount of x-ray radiation with a

detector composed of a material with higher atomic number (for example CdZnTe that has

Zeff = 50). For the typical value of the electric field used in a-Se devices (10 V/μm) the

value of is about 45 eV when for polycrystalline mercuric iodide (poly-HgI2) and

polycrystalline Cadmium Zinc Telluride (poly-CdZnTe) the value of is typically

5-6 eV.

4.4.2. Polycrystalline Cadmium Telluride (poly-CdTe)

CdTe has a moderate atomic number (Zeff ~ 50) and a low ~ 4.5 eV. As a consequence,

it is highly efficient to absorb the incident x-ray radiation and convert it to charge carriers.

As opposed to a-Se where both electrons and holes are mobile, in CdTe only electrons are

mobile. The major disadvantages of CdTe is the relatively high dark current which is

of the order of ~ 10 nA/cm2 and the high substrate and annealing temperatures

(180-190 oC) required to deposit large area polycrystalline CdTe layers by vacuum

deposition techniques. The latter can cause damage in the electronics which lie beneath

the photoconductor’s layer.

4.4.3. Polycrystalline Cadmium Zinc Telluride (poly-CdZnTe)

CdZnTe (<10% Zn) polycrystalline film has been used as a photoconductor layer in x-ray

AMFPI. Introduction of Zn into the CdTe lattice increases the bandgap, decreases

conductivity and hence largely reduces dark current. Hole mobility in CdZnTe decreases

32


with increasing Zn concentration whereas electron mobility remains nearly constant.

Furthermore, addition of Zn into CdTe increases lattice defects and hence reduces carrier

lifetimes. The poly-CdZnTe has a lower crystal density resulting in lower x-ray sensitivity

than its single crystal counterpart. Furthermore, for a detector of given thickness, the x-ray

sensitivity in CdZnTe detectors is lower than in CdTe detectors. Nevertheless, the CdZnTe

detectors show a better signal to noise ratio and hence give better detective quantum

efficiency (DQE). The measured sensitivities are higher than other direct conversion

sensors (e.g. a-Se) and the results are encouraging.

Although CdZnTe can be deposited on large areas, direct conversion AMFPI of

only 7.7×7.7 cm2 (512 × 512 pixels) from a polycrystalline CdZnTe has been

demonstrated. The CdZnTe layer thickness varies from 200-500 μm. Temporal lag and

nonuniform response were noticeable in early CdZnTe sensors which are attributed to

large and nonuniform grain sizes.

4.4.4. Polycrystalline Lead Oxide (poly-PbO)

Direct conversion flat panel X-ray imagers of 18 × 20 cm2 (1080 × 960 pixels) from a

poly-PbO with film thickness of ~300 μm have been demonstrated (Simon et al 2004).

One advantage of PbO over other x-ray photoconductors is the absence of heavy element

K-edges for the entire diagnostic energy range up to 88 keV, which suppresses additional

noise and blurring due to the K-fluorescence. PbO has ~8 eV, density 4.8 g/cm3,

energy gap Eg=1.9 eV and resistivity in the range 7-10 x 1012 Ω-cm (Simon et al 2004).

PbO photoconductive polycrystalline layers are prepared by thermal evaporation in a

vacuum chamber at a substrate temperature of ~100°C. The dark current in PbO sensors is

~40 pA/mm2 at an electric field of 3 V/μm (Simon et al 2004). PbO reacts with the air and

this leads to an increase in dark current and a decrease in x-ray sensitivity. Additionally,

thick PbO layers degrade from prolonged x-ray exposure.

4.4.5. Polycrystalline Mercuric Iodide (poly-HgI2)

Polycrystalline HgI2 layers can be prepared by both physical vapor deposition (PVD) and

screen printing (SP) from a slurry of HgI2 crystal using a wet particle-in-binder process

(Street et al 2002). Direct conversion x-ray AMFPI of 20 × 25 cm2 (1536 × 1920 pixels)

and 5 × 5 cm2 (512 × 512 pixels) size have been demonstrated using PVD and SP poly-

33


HgI2 layer, respectively (Street et al 2002, Zentai et al 2004). HgI2 has ~ 5 eV, density

6.3 g/cm3, energy gap Eg= 2.1 eV and resistivity ~ 4 x 1013 Ω-cm.

HgI2 tends to chemically react with various metals and hence a thin blocking layer

(typically ~1 μm layer of insulating polymer) is used between the HgI2 layer and the pixel

electrodes to prevent the reaction and also to reduce the dark current. The HgI2 layer

thickness varies from 100-400 μm.

The dark current of HgI2 imagers increases superlinearly with the applied bias

voltage. The dark current of a PVD HgI2 detector strongly depends on the operating

temperature (it increases by a factor of approximately two for each 6°C of temperature

rise). It is reported (Zentai et al 2004) that the dark current varies from ~ 2 pA/mm2 at

10°C to ~ 180 pA/mm2 at the 35°C for an applied electric field of 0.95 V/μm. On the

other hand, the dark current in the SP sample is an order of magnitude smaller than in

PVD sample and more stable against temperature variation. The only disadvantage of SP

detectors is that they show ~2−4 times less sensitivity compared to PVD detectors.

Electrons have much longer ranges than holes in HgI2. Furthermore, it is reported

that HgI2 image detectors with smaller grain sizes show good sensitivity and also an

acceptable uniform response. As reported in the literature, poly-HgI2 imagers show

excellent sensitivity, good resolution, and acceptable dark current, homogeneity and lag

characteristics, which make this material a good candidate for diagnostic x-ray image

detectors.

4.4.6. Polycrystalline Lead Iodide (poly-PbI2)

PbI2 photoconductive polycrystalline layers are prepared by PVD at a substrate

temperature of 200 to 230°C. Direct conversion AMFPI of 20 × 25 cm2 size (1536 × 1920

pixels) have been demonstrated using PVD polycrystalline PbI2 layer (Zentai et al 2003).

PbI2 coating thickness varies from 60-250 μm. PbI2 has ~ 5 eV, energy gap Eg=2.3 eV

and resistivity in the range 1011-1012 Ω-cm.

PbI2 detectors have a very long image lag decay time that depends on the exposure

history. The dark current of PbI2 imagers increases sublinearly with the applied bias

voltage and it is in the range 10-50 pA/mm2 at an electric field of 0.5 V/μm. Furthermore,

it is much higher than that of PVD HgI2 detectors, making it unsuitable for long exposure

time applications. The resolution of PbI2 imagers is acceptable but slightly less than that of

HgI2 imagers. Also, the x-ray sensitivity of PbI2 imagers is lower than that of HgI2

34


imagers. The pixel to pixel sensitivity variation in PbI2 imagers is substantially low. Its

dark conductivity is larger than that for a-Se (Kasap and Rowlands 2000).

4.5. Table of material properties

In table 4.1 some of the material properties of a-Se, a-As2Se3, GaSe, GaAs, Ge, CdTe,

CdZnTe, Cd0.8Zn0.2Te, ZnTe, PbO, TlBr, PbI2 and HgI2 are presented.

Table 4.1. Some materials properties of potential x-ray photoconductors for digital

mammography. The data are obtained from Kasap and Rowlands (2000, 2002a, 2002b),

Kasap 1991, Bencivelli et al (1991).

35

Photoconductor /state

Eg (eV)

W (eV)

Density (g/cm3)

Resistivity (Ω cm)

a-Se Amorphous 2.3 45

(at 10 V/μm) 4.3 1014-1015

a-As2Se3 Amorphous 1.8 ~ 4.46 4.55 1012

GaSe Crystalline 2 6.3 4.6

GaAs Crystalline 1.42 6.3 5.31 107

Ge Crystalline 0.7 1.5 5.32 102

CdTe Polycrystalline 1.5 4.65 6.06 109

CdZnTe, Cd0.8Zn0.2Te Polycrystalline

1.7 5 5.8 1011

ZnTe Crystalline 2.26 7 6.34

PbO Polycrystalline 2.7 8-20 9.8 7-10 x 1012

TlBr Crystalline 2.7 6.5 7.5 1012

PbI2 Polycrystalline 2.3 5 6.1 1011-1012

HgI2 Polycrystalline 2.1 4.1 6.3 4 x 1013

CHAPTER 5: PHYSICS OF IMAGE FORMATION

CHAPTER 5

PHYSICS OF IMAGE FORMATION

36


5.1. Introduction

The x-ray-matter interactions produce charges (i.e. electrons) that as they drift inside the

photoconductor interact with the material giving birth to secondary charged particles. The

final image is created from those charges that have not been recombined or trapped during

their drifting towards the collecting electrodes. This chapter describes the basic

physics of x-ray–matter interactions, of atomic deexcitation mechanisms and of electron

interactions. Furthermore, it discusses some aspects of the charge carrier transport inside

a-Se.

5.2. X-ray – matter interactions

In the mammographic energy range (photon energies smaller than 40 keV) three are the

types of x-ray-matter interaction:

A. Coherent (Rayleigh) scattering.

B. Incoherent (Compton) scattering.

C. Photoelectric absorption.

5.2.1. Coherent (Rayleigh) scattering

Rayleigh scattering is a process in which the energy of the initial photon is not converted

to kinetic energy of another particle but it is all scattered.

Using classical physics, one can derive the differential cross section of scattering of

a photon from a free electron (Thomson scattering) as:

(5.1)

where ro is the classical radius of the electron, θ is the angle between the initial trajectory

of the electromagnetic wave and the new one after the scattering from the electron, and

dσο/dΩ is the differential cross section per electron of the classical scattering that gives the

fraction of the incident energy that is scattered by the electron into the solid angle dΩ=1/r 2

or the fractional number of photons scattered into unit solid angle at angle θ.

Suppose now that the photon of energy E passes over an atom. Since a photon is an

electromagnetic wave, its oscillating electric field sets the electrons of the atom in a

momentary vibration. These oscillating electrons emit radiation of the same energy E as

the incident radiation. The scattered waves from electrons combine with each other to

form the scattered photon. The differential cross section for Rayleigh scattering is given

as:

37


(5.2)

where F( , Z) is the so called ‘atomic form factor’, which is the probability that all the

electrons take up recoil momentum without absorbing any energy, and is related to the

momentum transfer during the collision given by:

(Å-1) (5.3)

where mc2 is the electron rest energy.

5.2.2. Incoherent (Compton) scattering

In Compton scattering a photon with energy E collides with an atomic electron, transfers

some of its energy and momentum to the electron and is being deflected at an angle θ,

with respect to its initial direction, with energy given by:

(5.4)

The differential cross section of incoherent scattering including electron binding effects

can be given as the product of the Klein-Nishina differential cross section dσKN/dΩ

(for Compton collision between a photon and a free electron) and the incoherent scattering

function of an atom S( , Z). The latter represents the probability that an atom will be

raised to any excited or ionized state when a photon imparts a recoil momentum to an

atomic electron. Therefore the differential cross section of incoherent scattering is given

by:

(5.5)

The relationship between the electron scattering angle θe and the photon scattering angle θ,

derived from angle calculations in the scattering process, is given by:

(5.6)

5.2.3. Photoelectric absorption

In the photoelectric effect the incident photon is completely absorbed by the atom that

ejects a photoelectron. The more tightly bound electrons are the important ones in

38


bringing about photoelectric absorption and the maximum absorption occurs when the

photon has just enough energy to eject the bound electron. The photoelectron is ejected

with energy Ee given by:

Ee=E-Bs (5.7)

where Bs is the binding energy of the electron in the ‘s’ shell. To a first approximation, at

low energies the emission is entirely due to the electric vector of the incident wave acting

on the electron. Neglecting relativity and spin corrections, the number of photoelectrons

per solid angle is (Davisson and Evans 1952):

(5.8)

where β=υ/c with υ being the electron velocity, and are the polar and azimuthal

angles of the photoelectron with respect to the direction of the incident photon.

5.3. Atomic deexcitation

The atom after having interacted with photoelectric effect deexcites releasing energy.

During a deexcitation sequence, the vacancy initially created in an inner shell migrates to

outer shells with radiative and non-radiative atomic transitions. The radiative transitions

are the emission of fluorescent photons while the non-radiative transitions the ejection of

Auger or Coster-Kronig (CK) electrons.

Figure 5.1. illustrates schematically the three types of atomic deexcitation. In a

radiative transition (figure 5.1(a)), the vacancy in a shell is filled in with an electron that

jumps from a higher shell or subshell with the concomitant emission of electromagnetic

radiation (fluorescent photon). The inter-subshell fluorescent transitions (e.g. the emission

of XiXj photon in figure 5.1(a)) have almost negligible probability of

occurrence. The fluorescent photons are isotropically emitted with energy E fl = Bi-Bj,

where Bi and Bj are the binding energies of the shells i and j involved in the transition.

The Auger electron emission (figure 5.1(b)) occurs when the vacancy in a shell is

filled in with an electron that jumps from a higher shell and another electron is ejected

from that shell (e.g. the XYY Auger in figure 5.1(b)) or a higher one (e.g. the XYZ Auger

in figure 5.1(b)). On the other hand, the CK electron emission (figure 5.1(c)) occurs when

the vacancy in a shell is filled in with an electron that jumps from a higher subshell of the

same shell and another electron is ejected from a higher shell (e.g. the X iXjY CK in figure

5.1(c)). As it is the case for the fluorescent photons, the Auger and the CK electrons are

39

isotropically ejected, while their energy is EAug/CK = Bi-Bj-Bk, where Bi, Bj and Bk are the

binding energies of the shells i, j and k involved in the transition.


Figure 5.1. The three types of atomic deexcitation. (a) Fluorescent photon emission,

(b) Auger electron ejection and (c) CK electron ejection.

40

X-shell

Xj-subshell

Xi-subshell

Y-shell

XiXj -photon

XiY -photon

X-shell

Y-shell

Z-shell

XYY-Auger

XYZ-Auger

X-shell

Xj-subshell

Xi-subshell

Y-shell

Z-shellXiXjY-CK

XiXjZ-CK

(a)

(b)

(c)


5.4. Electron interactions

Five are the types of electron-matter interaction:

A. Emission of Bremsstrahlung radiation.

B. Phonon interactions.

C. Elastic scattering.

D. Inelastic scattering.

E. Collective oscillations (plasma waves).

The emission of Bremsstrahlung radiation takes place at high energies and high Z

materials. Thus, in the mammographic energy range this effect can be ignored. The

phonon interactions, alternatively called ‘inelastic collisions of low energy electrons with

phonons’, are not well understood and take place for electron energies lower than 50 eV

(Fourkal et al 2001). Consequently this kind of interaction is not discussed.

5.4.1. Elastic scattering

Elastic interactions are those in which the initial and final quantum states of the target

atom are the same. During an elastic collision, there is a certain energy transfer from the

projectile to the target, which causes the recoil of the latter. Because of the large mass of

the target, the average energy lost by the particle is a very small fraction of its initial

energy and is usually neglected. This is equivalent to assuming that the target has an

infinite mass and does not recoil. For a wide energy range (a few hundred eV to ~1GeV),

elastic collisions can be described as scattering of the projectile by the electrostatic field of

the target.

In the independent atoms approximation it is assumed that the target atoms are

independent, neutral and at rest. To account for the effect of the finite size of the nucleus

on the elastic differential cross section (DCS) (which is appreciable only for projectiles

with energies larger than a few MeV) the nucleus can be represented as a uniformly

charged sphere of radius:

Rnuc=1.05 x 10-15 Aw1\3 (5.9)

where Aw is the atomic mass in g/mol. The electric field that the projectile encounters is

that of the nucleus and of the electron cloud. The electrostatic potential of the target atom

is (Salvat et al 2003):

41


(5.10a)

where ρ is the electron cloud density and φnuc is the potential of the nucleus given by:

(5.10b)

In the static-field approximation (Mott and Massey 1965, Walker 1971) the DCS for

elastic scattering is obtained by solving the partial wave expanded Dirac equation for the

motion of the projectile in the field of the atom. The interaction energy is:

V(r)= -eφ(r)+Vex(r) (5.11)

with Vex(r) being the local approximation to the exchange interaction between the incident

electron and the atomic electrons. It is assumed that the electron has randomly oriented

spin. Therefore, the effect of elastic interactions can be described as a deflection of the

projectile trajectory, characterized by the polar θ and azimuthal φ angles. For a central

field, the angular distribution of singly scattered electrons is axially symmetric about the

direction of incidence (independent of φ). The DCS (per unit solid angle) for elastic

scattering of a projectile with kinetic energy E, into the solid angle element dΩ about the

direction (θ,φ) is given by (Walker 1971):

(5.12a)

where f(θ) is the direct scattering amplitude given by:

(5.12b)

with δl+ and δl- being the phase shifts and Pl(cosθ) the associated Legendre polynomials,

and g(θ) is the spin flip scattering amplitude given by:

(5.12c)

with Pl1(cosθ) being the associated Legendre functions. In the above equations k is the

wave number of the projectile given by:

(5.13)

42


For calculation simplicity, the Rutherford elastic scattering differential cross section is

usually used and it is given by (Shimizu and Ze-Jun 1992):

(5.14)

where ζ is the screening parameter. Nevertheless, the above DCS is not a realistic

approximation.

5.4.2. Inelastic scattering

Inelastic collisions are the dominant energy loss mechanism for electrons with

intermediate and low energies. They are interactions that produce electronic excitations

and ionizations in the medium.

5.4.2.1. The development of a theory for inelastic collisions

The theory of energy loss of fast charged particles (meaning that their kinetic energy is

much bigger than the kinetic energy of the atomic electrons) caused by their inelastic

collisions with atoms was established by Niels Bohr (1913) through a semiclassical

treatment. In this approach, collisions are classified according to their impact parameter b,

which is, roughly, the distance of closest approach of the incident particle to the center of

the atom. The theory has been developed for the case of separate atoms (gases).

Bethe (1930, 1932) formulated the theoretical expression for inelastic scattering

stopping power of electrons in gases (free atoms and molecules) at a quantum mechanical

base. He classified the collisions according to their momentum transfer q, which is

observable in contrast to b. The momentum transfer q is a function of the energy transfer

W of the incident particle to the atom and of the deflection angle θ experienced by the

incident particle. The theory is based on the so called First Order (Plane Wave) Born

Approximation:

First Order (Plane Wave) Born Approximation: The scattering field is considered to be a

perturbation (to first order). That is, the interaction factor V (a Coulomb potential/field)

between the particle and the atom, is calculated in the lowest order.

Behte has written the inelastic scattering cross section σn in a differential form with

respect to the final momentum of the incident particle as (Fano 1963):

43


(5.15)

where p, E, and u are the initial momentum, kinetic energy and velocity of the incident

particle in respect, , and the corresponding values after the collision and En the

energy of the final stationary state of the atom (whose initial energy was E0=0). The δ

function imposes energy conservation. Equation (5.15) can be rewritten as:

(5.16)

Bethe’s calculations on the stopping power are based on the continuous slowing down

approximation (CSDA). He included the contribution of all possible atomic excitation

processes to the energy loss in a factor called the mean ionization energy J. Thus, the

average energy loss ΔE of a penetrating electron for a given path length S is given by:

(5.17a)

with

(5.17b)

where E is the energy of the incident electron, NA is the Avogadro number, ρ is the density

and A is the atomic weight. Berger and Setzer (1964) developed an empirical formula for

calculating J:

(5.18)

Nevertheless, the Bethe stopping power (5.17b) has two basic problems:

1. It is not valid for energies lower than J because the logarithmic term becomes

negative for energies < J/1.66

2. The CSDA does not allow the secondary cascade generation and simple excitation

processes to be described.

5.4.2.2. Recoil energy

Figure 5.2 presents a simplified schematic of an inelastic collision.

44

(E, p)

(E’, p’) θ

θr

q=p-p’


Figure 5.2. A simplified schematic of an inelastic collision of an electron with initial

energy and momentum (E,p) with an atom, during which the electron transfers momentum

q to the atom and is scattered at angle θ with (E’, p’).

Suppose an electron with energy and momentum (E, p) that interacts with an

initially free and at rest electron. The primary electron transfers a momentum q to the

target electron and loses an energy W. In this situation a given momentum transfer q

results in a unique value of energy transfer W. Thus, the incident electron acquires

momentum p’=p-q, energy E’=E-W and is deflected at angle θ. The target electron

acquires energy Q (recoil energy), momentum q and is deflected at angle θr. It is obvious

that Q=W and Q is given by:

(relativistic) (5.19a)

Q= (non-relativistic) (5.19b)

Suppose now that the electron interacts with the electrons in an atom and that it transfers

momentum q to the atom as a whole. Due to the fact that the atomic electrons, which are

responsible for internal atomic excitations, are bound and move around the nucleus, in this

case there is not a unique correspondence between q and W, θ but W and θ can acquire

different values depending on the atomic excitation. That is, in the case of an inelastic

collision of an electron with an atom, both the energy loss W and the scattering angle θ of

the projectile are stochastic quantities that are defined through certain probability density

functions. For a better handling of calculations, it is more convenient to adopt the recoil

energy Q instead of θ, using equation (5.19) and:

(5.20)

5.4.2.3. Bethe’s theory revisited

Inokuti (1971) has rewritten equation (5.16) for the non-relativistic case as:

(5.21)

45


In this equation:

o K is related to the momentum transfer q= K= (k-k’)=p-p’.

o is the recoil energy.

o The factor is the Rutherford cross section for the

scattering of a particle with charge z0e (z0=1 for an electron) by a free and initially

stationary e, which upon the collision receives recoil energy between Q and Q+dQ.

o εn(K) is a matrix element related to q. The factor accounts for the fact

that if the incident particle transfers momentum to an atom as a whole, there is not a

unique correspondence between momentum transfer q and energy transfer W. It gives

the conditional probability that the atom makes a transition to a particular excited state

n upon receiving a momentum transfer q. It is called the inelastic scattering form

factor.

5.4.2.4. Generalized Oscillator Strength -Optical Oscillator Strength

The concept of oscillator strength stems from the late 19 th century model of the electrical

and optical behavior of matter. Electrons were supposed to lie at equilibrium positions

within atoms and to react elastically to weak disturbances. Thus, they would perform

forced oscillations when exposed to electromagnetic radiation. The amplitude and phase

of these oscillations would depend on the characteristic (angular) frequency of free

oscillation of each electron ωs, on its weak damping constant γs, and on the radiation

frequency ω. In particular, if the disturbance is an electromagnetic wave with an

oscillating electric field F = Fo exp(-iωt) and it is directed along z, an atomic electron,

which is also subject to an elastic force –mωs2z and to a frictional force –mγsdz/dt,

experiences a displacement z from its equilibrium position given by:

(5.22)

Therefore the disturbed electron and the atomic nucleus form an oscillating electric dipole

which has a dipole moment ez. The polarizability of this dipole a(ω), that is the dipole

moment ez per unit field strength, is given as:

(5.23)

46


The complex character of α(ω) serves to represent by a single number the phase lag and

the magnitude of the displacement z with respect to the field oscillation. If we now

consider the interaction of incident electromagnetic wave with all the atomic electrons,

then if ns is the number of electrons at each state ‘s’, we have fs (=ns) number of oscillators

at each state ‘s’ that take part in the interaction. The number fs is called the generalized

oscillator strength (GOS), and is the number of dipole oscillators of natural frequency ωs,

or else the number of electrons at state ‘s’.

In the quantum mechanical treatment, the analogue of the classical oscillator strength

fs is a function g(k, n) which is proportional to the probability of an electron passing from

state |k> to |n>. Therefore for a one electron atom the sum of oscillator strengths over all

the possible n states is 1. Thus for an atom of Z electrons the same sum is equal to Z. This

is an initial description of the so called ‘Bethe sum rule’ later discussed in more detail.

Coming back to equation (5.21) one can define the GOS as (Inokuti 1971):

(5.24a)

or

(5.24b)

with . The optical oscillator strength

(OOS) is defined as:

(5.25)

When one deals with excitation to continua (i.e. with ionization) the excitation energy En

is not a discrete variable, but it is a continuous variable that takes all real values greater

than the first ionization threshold. Then the inelastic cross section for excitation to

continuum states between and +d is . Thus is the differential inelastic

cross section per . In this case the GOS is defined as (Inokuti 1971):

(5.26)

The final continuum state is specified by and a set Ω of all the other quantum numbers

(i.e. angular momentum or direction of atomic electron ejection). The GOS describes the

atom. It is difficult to be evaluated theoretically, because a sufficiently accurate

47


eigenfunction of an atomic or molecular system in its ground state and especially in its

excited states is seldom available. Atomic Hydrogen and the free electron gas are the only

systems for which the GOS is known for every transition.

The effect of individual inelastic collisions on the projectile is completely specified

by giving the energy loss W and the polar θ and azimuthal φ scattering angles

(W, θ, φ) or (W, Q, φ). For media with randomly oriented atoms the DCS for inelastic

collisions is independent of the azimuthal angle φ. Thus the important parameters are the

recoil energy Q and the energy loss W. Therefore, it is more convenient to work in the so

called (Q, W) representation.

In (Q, W) representation the GOS and OOS are defines as:

(5.27)

The OOS is closely related to the photoelectric cross section for photons with energy W.

That is, as long as the wavelength of the incident particle is sufficiently large compared to

the atomic size (dipole approximation), the OOS is proportional to the cross section for

photoelectric absorption of a photon with energy W by the atom. The knowledge of GOS

does not suffice to describe the energy spectrum and angular distribution of secondary

knock on electrons (δ-rays).

5.4.2.5. Bethe surface- Bethe sum rule

The plot of GOS on (Q, W) plane is called the Bethe surface. The physics of inelastic

collisions is largely determined by a few global features of the Bethe surface:

o Limit Q-> 0: In this limit the GOS reduces to OOS.

o Limit of very large Q: In this limit, the binding and momentum distribution of the

target electrons have a small effect on the interaction. Thus, in the large Q region, the

target electrons behave as if they were free and at rest and consequently the GOS

reduces to a ridge along the line W=Q which was named the Bethe Ridge.

For the discrete case the Bethe sum rule is (Inokuti 1971):

(5.28)

whereas for the continuum case and in the (Q, W) representation (Salvat et al 2003):

(5.29)

48


The Bethe sum rule roughly says that the average of the energy transfer to the atom over

all modes of internal excitation for a given Q should be the same as the energy transfer to

Z free electrons.

5.4.2.6. The differential inelastic scattering cross section

Fano (1963) using equations (5.21) and (5.24) calculated the differential cross section for

inelastic scattering of a charged particle from an isolated atom (independent’s atom

approximation). This cross section can be calculated adequately to lowest order in the

particle-atom interaction, or else in the low-Q approximation. The obtained differential

cross section can be written for the continuum case as (Salvat et al 2003):

(5.30)

where β=u/c with u being the velocity of the incident electron and θr is the angle between

the initial momentum of the projectile and the momentum transfer which is given by:

(5.31)

In (5.30) the first term in the curled bracket deals with the longitudinal interactions. These

are related to the Coulomb interaction between the incident electron and the atom, an

interaction that is parallel (longitudinal) to the momentum transfer q. The second term in

the curled bracket is related to the transverse interactions. The mechanism of transverse

excitations by a fast particle is electromagnetic and as such becomes important only when

the particle approaches the light velocity or when the energy taken up by an electron in the

medium is itself relativistic. This interaction is also known as interaction through virtual

photons and is called transverse because the photon fields are perpendicular to q.

The differential inelastic scattering cross section for dense media (solids) can be

obtained from a semiclassical treatment in which the medium is considered as a dielectric,

characterized by a complex dielectric function , which depends on the wave

number k and the frequency ω. In the classical picture, the electric field of the projectile

polarizes the medium producing an induced electric field that causes the slowing down of

the projectile. The dielectric function relates the Fourier components of the total

(projectile’s and induced) electric field and the external electric potentials. The

momentum transfer can be defined as and the energy transfer as W= and

49


therefore . The DCS obtained from the dielectric and quantum treatments are

consistent (the former reduces to the latter for a low-density material) if one assumes the

identity:

(5.32)

where Ωp is the plasma energy of a free-electron gas given as:

(5.33)

where N is the electron density of the medium. The differential inelastic scattering cross

section for dense media is given by (Salvat et al 2003):

(5.3

4)

The factor D(Q, W) accounts for the so-called density effect correction (Sternheimer

1952). The origin of this term is the polarizability of the medium, which screens the

distant transverse interactions causing a net reduction of their contribution to the stopping

power.

5.4.2.7. Secondary electron emission (δ-rays)

After the collision of the primary electron with an electron of an atom, the energy W lost

by the primary electron is transferred to the secondary electron and if this electron is

ejected from an inner shell it acquires energy Es=W-Bi, where Bi is the binding energy of

the particular inner shell, or if it is ejected from an outer shell its energy is E s=W. If it is

assumed that the atomic electron is initially at rest, then it is ejected at the direction of

momentum transfer q and therefore its polar ejection angle is given by (5.31). Since the

momentum transfer lies on the plane formed by the initial and final momenta of the

primary electron (scattering plane), the azimuthal emission angle φs of the secondary

electron is φs=φ+π, where φ is the azimuthal angle of the scattered projectile.

5.4.3. Collective description of electron interactions

When an electron transverses a medium, it is likely the energy transferred to the medium

to induce collective oscillations of electrons (plasma waves). This kind of interaction is

possible in materials like a-Se (Fourkal et al 2001). A plasma wave is possible to decay

into many electron-holes pairs.

50


The theoretical formulation of the collective oscillations of electrons has been

formulated by Pines and Bohm (1952). The theory developed deals with the physical

picture of the behavior of the electrons in a dense electron gas. In a dense electron gas, the

particles interact strongly because of the long range of the Coulomb force. In fact, each

particle interacts simultaneously with all the other particles.

Suppose a particle that moves inside a dense electron gas with velocity u0. If its

velocity is smaller than the mean thermal speed of electrons in the gas, the electrons

respond in such a way that when a steady state is established, the field of the particle is

screened out within a distance λD which is the Debye length given by:

(5.35)

where kB is the Boltzmann’s constant, n the electron density and T the temperature. The

picture is as follows: suppose an electron at position xi having velocity smaller than the

thermal speed. The particle is surrounded by a comoving cloud in which the electron

density is reduced below the average. The cloud is elliptical, being shortened in the

direction of particle motion by a specific ratio. The comoving cloud represents a region

from which electrons have been displaced by the repulsive Coulomb potential of the ith

electron. Most of the electron cloud is located within a distance ~ λD. The electric field of

the ith electron for r>λD is negligible (screening).

If the particle has u0 bigger than the mean thermal speed, the field of the particle

continues to be screened, but also a new phenomenon appears: the excitation of a wake

trailing behind the particle consisting of collective oscillations that carry energy away

from the particle. The energy loss to the collective oscillations is of the same order of

magnitude as the loss caused by short-range Coulomb collisions with the individual

particles. The energy lost per unit distance to the collective oscillations is given by:

(5.36)

where <v2>Av is the average value of the mean thermal velocity of electrons in solid and

E0=mu02/2. The mean free path an electron travels for the emission of an energy quantum

to collective oscillations is:

(5.37)

where and ωp is the plasma frequency. The induction of collective oscillations

from a passing electron does no produce angular deflection to the electron.

51


5.5. Charge carrier transport inside a-Se

As it was mentioned, the x-ray matter interactions produce electrons (primary electrons).

These electrons as they travel inside the solid cause ionizations along their tracks and

hence the creation of electron-hole pairs. For many semiconductors the energy

required to create an electron-hole pair has been shown to depend on the energy bandgap

Eg via the Klein rule (Klein 1968):

(5.38)

The phonon term Ephonon accounts for energy losses (like phonon production) and is

typically very small (~0.5 eV). Thus, practically . Que and Rowlands (1995b)

have shown that for the case of amorphous semiconductors Klein’s rule is written as:

(5.39)

For the case of a-Se (Eg~2.2. eV) one would expect ~5 eV. Nevertheless, experimental

measurements have found that >>5 eV. In addition, experiments show that

decreases with increasing the electric field (Rowlands et al 1992) and that it has an energy

dependence (Mah et al 1998). These observations imply that the charge carriers inside

a-Se are subject to recombination and trapping. When an electron-hole pair is created

inside a-Se it may:

i. Recombine with the other half of the same pair before they are separated (geminate

recombination).

ii. Separate to recombine with other electrons or holes in the same electron track. This

is the so-called columnar recombination, so named because a column of ionization

forms around the electron track.

iii. Separate, escape from the track to recombine in the bulk of the a-Se with electrons or

holes from other tracks (bulk recombination).

iv. Separate, escape from the track to become trapped in the photoconductor layer (bulk

trapping).

v. Separate, escape the track, avoid trapping and reach one of the surfaces of the a-Se

layer.

Haugen et al (1999) showed that bulk recombination inside a-Se is negligible.

Furthermore, Kasap et al (2004) showed that only drifting holes recombine with trapped

electrons, following Langevin recombination with coefficient:

52


(5.40)

where μh is the hole drift mobility, εoεr is the permittivity of a-Se, whereas the

recombination of drifting electrons with trapped holes is negligible.

5.5.1. Geminate (Onsager) recombination

The theory for geminate (initial) recombination of ions was formulated by Onsager

(1938). Later on, Pai and Enck (1975) used Onsager theory to investigate the

photogeneration process inside a-Se in the optical regime.

Each absorbed optical photon creates one electron-hole pair in a-Se. The excess

kinetic energy Ek carried by the electron or hole is not sufficient to generate secondary

electrons and holes, and is presumed to be dissipated by exciting phonons. The process by

which the electron-hole pair loses excess energy and reaches an equilibrium state is called

the ‘thermalization process’. After the electron-hole pair is thermalized, the electron and

hole are separated by a distance r and at an angle θ with the applied field F. According to

Onsager theory, such a thermalized pair can either recombine (geminate recombination) or

escape their mutual Coulomb attraction and separate into a free electron and a free hole.

The probability of escaping geminate recombination is:

(5.41)

where and ε is the relative dielectric constant, kB the

Boltzann’s constant and T the temperature. The probability p(r, θ, F) increases as r, θ, F

increase. The distribution of r in the electron-hole pairs is given by (Pai and Enck 1975):

(5.42)

where ro is characteristic thermalization length determined experimentally. If the

photogeneration efficiency n is defined as the fraction of electron-hole pairs which do not

recombine relative to all electron-hole pairs created, then:

53


(5.43)

where and Il are the modified Bessel functions. The

photogeneration efficiency increases as ε, ro and T increase, whereas n and depend on

the energy of incident photon Eph , and F, T. Knights and Davis (1975) assumed that

during the thermalization process the motion of the carriers is diffusive and the rate of

energy dissipation to phonons is hνp2. Thus, they have calculated the thermalization

distance ro as:

(5.44)

where t is the thermalization time, D is the diffusion constant and hν is the incident

photon’s energy.

Que and Rowlands (1995b) have extended the Onsager theory into the x-ray regime.

At high electric fields (i.e. 10 V/μm) g(r,θ)=g(r) and the distribution g(r) is given as:

(5.45)

where is the distribution of the kinetic energies Ek of the electron-

hole pairs and Ec=2.067 eV (for a-Se). The photogeneration efficiency is now given as:

(5.46)

where . To a first order with respect to F (5.46) is written as:

(5.47)

The thermalization distance r is obtained from:

(5.48)

54


where and for a-Se: μ = μh = 0.14 cm2/Vs and hνp2=15 meV. The n and do

not depend on Eph but still depend on F, T.

5.5.2. Columnar recombination

Jaffe (1913) formulated the theory for columnar recombination. As mentioned earlier, in

columnar recombination the electron-hole pairs separate and the released charges

recombine with other electrons or holes within the same electron track. Jaffe’s

assumptions are:

i. Only one electron track is taken into consideration. The time needed for the creation

of the track is much smaller than the recombination time of the carriers.

ii. The holes and electrons are ejected from the central axis of the column in opposite

directions.

iii. The mobilities of electrons and holes are equal.

iv. Diffusion, external field and recombination are taken into account. The electric field

of electrons and holes is neglected.

Figure 5.3. The geometry that Jaffe used in his calculations for columnar recombination.

v. N particles (charges) are created homogeneously per unit length of the track,

whereas the initial distribution around the center of the column is Gaussian:

(5.49)

with b being the column’s initial radius.

Jaffe calculated, as a function of time, the spatial distribution of charges in the

column n(r, t) as well as the fraction of electrons that escape columnar recombination:

(i) in the absence of any applied field F, (ii) when F is parallel with the track, (iii) when F

55

x

z

y

Penetrating electron’s track

Formed charge column


is perpendicular to the track and (iv) when F makes an angle φ with the track. The

geometry used is shown in figure 5.3.

5.5.2.1. Absence of electric field

The fraction of electrons that escape recombination are assumed to be those that travel a

distance R from the central axis of the column. Hence:

(5.50)

with and No the initial number of charges, α

the recombination coefficient, D the diffusion coefficient and b the column’s initial radius.

In the absence of an applied field the number of electrons that escape recombination is

negligible.

5.5.2.2. Field parallel to the column

If one assumes that initially No charges were created in length d, then the fraction of

charges that escape columnar recombination is:

(5.51)

where μ is the charge mobility and and

5.5.2.3. Field perpendicular to the column

In this case the fraction of electrons that escape recombination is:

(5.52)

with

, , ,

5.5.2.4. Field at an angle φ with the track

56


The fraction of electrons that escape recombination is:

(5.53)

with . It is seen that for small F, , whereas for

intermediate and large F, .

57

CHAPTER 6: MONTE CARLO SIMULATION

CHAPTER 6

MONTE CARLO SIMULATION

58


6.1. Introduction

The Monte Carlo method has long been recognized as a powerful technique for

performing certain calculations, generally those too complicated for a more classical

approach. Since the use of high-speed computers became widespread in the 1950s, a great

deal of theoretical investigation has been undertaken and practical experience has been

gained in the Monte Carlo approach.

Historically, a primitive Monte Carlo method was first used by Captain Fox to

determine π in 1873. During World War II, Von Neumann and Ulam introduced the term

Monte Carlo as a code word for the secret work at Los Alamos. It was suggested by the

gambling casinos at the city of Monte Carlo in Monaco. The Monte Carlo method was

then applied to problems related to the atomic bomb. After 1944, Fermi and Ulam used the

method to study the Schrödinger equation in quantum mechanics and Goldberger to study

nuclear fusion (1948). During the period 1948-1952, Wilson R R, Berger M and

McCracken, used Monte Carlo techniques to conduct research in x-rays and a-rays

showers. Soon after the initiation of computers in Monte Carlo calculations, the method

was used to evaluate complex multidimensional integrals and to solve certain integral

equations, occurring in physics, which were not amenable to analytic solution.

The application of Monte Carlo methods is a procedure which can be considered as a

two input-one output problem as shown in figure 6.1. The two inputs are a large source of

high quality random numbers and a probability distribution which describes the

considered problem, whereas the output is the result of the random sampling of the

probability distribution.

Figure 6.1. General block diagram of a Monte Carlo procedure.

59

Random Numbers

Probability Distribution

Monte Carlo Results


In general, the primary components of the Monte Carlo method are the following:

Probability Distribution Functions (PDFs): the physical (or mathematical) system

must be described by a set of PDFs.

Random Number Generator: a source of random numbers uniformly distributed on

the unit interval must be available.

Sampling rule: a prescription for sampling from the specified PDFs, assuming the

availability of random numbers on the unit interval, must be given.

Scoring (or tallying): the outcomes must be accumulated into overall tallies or

scores for the quantities of interest.

Error estimation: an estimate of the statistical error (variance) as a function of the

number of trials and other quantities must be determined.

Variance Reduction Techniques: methods for reducing the variance in the

estimated solution to reduce the computational time for the Monte Carlo simulation.

Parallelization and vectorization: algorithms to allow Monte Carlo methods to be

implemented efficiently on advanced computer architectures.

In the next sections some aspects of the mathematical foundation for Monte Carlo

calculations as well as the Monte Carlo methods will be described. The information is

obtained from Morin et al (1988).

6.2. Random numbers-Random variables

The necessity to use random numbers emerged basically for three reasons:

i. Due to the need to study physical phenomena that their nature was random (e.g. the

thermionic emissions of electrons from a metal).

ii. Due to the fact that some physical problems were very expensive or very dangerous

to be studied from an experiment or there were no experimental data.

iii. Due to the fact that some phenomena because of their complex nature (e.g. the

random Brownian motion), were better to be studied using random numbers than

actually be studied analytically.

A random number is a particular value of a continuous variable uniformly distributed

on the unit interval, which together with others of its kind, meets certain conditions. A

high quality random number sequence is a long stream of numbers with the characteristic

that the occurrence of each number in the sequence is unpredictable and that the stream of

digits of the sequence passes certain tests which are designed to detect departures from

randomness. The quality of a supposedly random sequence of numbers can be established

60


only after a careful analysis aimed at discovering pattern in the sequence. The larger the

number of tests applied, and the higher the level of sophisticated of these tests, the higher

is the quality of a sequence which passes the test.

There are many random number sources, which are usually classified based on their

method of production into three categories: tables, physical sources and algorithms. The

latter mentioned, may appear to be a contradiction because an algorithm is a detailed set of

rules to obtain a specific output from a specific input. Such an algorithm is termed a

Random Number Generator (RNG), and its output is formally called a pseudorandom

number, reflecting its deterministic production.

One important disadvantage of the use of algorithmic random number generators, is

the fact that after a certain number of distinct elements have been produced, the sequence

begins to repeat. If the period of the sequence, that is the number of distinct digits, is large,

then the periodic behavior of a particular algorithmic random number generator is of no

practical importance. So, it is important to establish the fact that the period is sufficiently

large for the intended purpose of the generator. Consequently, the important

characteristics of a RNG is the long period and the uniformity, in the sense that equal

fractions of random numbers should fall into equal ‘areas’ in space.

Lehmer’s method is still the most commonly used for RNG. It is called the

multiplicative-linear-congruential method. Given a modulus M, a multiplier A, and a

starting value ξo (the seed), random numbers ξi are generated according to:

ξi=(Aξi-1+B)moduloM (6.1)

where B is a constant (Andreo 1991).

A random variable is a variable that can take on more than one value (generally a

continuous range of values) and for which any particular value that will be taken cannot de

predicted in advance. Even though the value of the variable is unpredictable, the

distribution of the variable may well be known. The formal definition of a random

variable is given as:

Random Variable: A function whose value is a real number determined by each element

in the sample space.

6.3. Probability Distribution Functions - Cumulative Distribution Functions

When modeling a physical system or a physical phenomenon using Monte Carlo

techniques, the first step that must be taken is the interpretation of this system or

61


phenomenon with a set of mathematical expressions. These expressions must include all

the physics describing the system (phenomenon) and can be derived either directly from

theory or using experimental results. From these expressions, a PDF or a set of PDFs is

made. Obviously, depending on the mathematical expressions used, PDFs can either have

their origins in experimental data or in a theoretical model. Modeling the physical process

by one or more PDFs, one can sample an ‘outcome’ from them (for example sample the

azimuthal angular distribution of photoelectrons in the photoelectric process). Thus, the

actual physical process is simulated. The PDF can be defined as:

Probability Distribution Function (PDF): The function f(x) is a probability density

function for the continuous random variable X, defined over the set of real numbers R, if

i. f(x) 0 for all x R

ii.

iii. P(a<X<b)=

where P(a<X<b) is the probability of X to take a value between a and b.

In some cases it is more convenient to use the cumulative distribution function (CDF)

defined as:

Cumulative Distribution Function (CDF): The cumulative distribution function F(x) of a

continuous random variable X with distribution function f(x) is given by:

F(x) = P(X x) =

(6.2)

As can be seen, F(x) is a monotonically non-decreasing function taking on values from

zero to one.

6.4. Sampling techniques

Once the PDF describing the physical system is known, the next step in a Monte Carlo

simulation is the sampling process. The sampling process is a procedure in which the PDF

is randomly sampled basically by two methods:

A. The Inversion Method.

B. The Rejection Method.

62


6.4.1. The Inversion Method

The inversion method is described by the fundamental inversion theorem:

Fundamental Inversion Theorem: Let X be a random variable with PDF f(x), cumulative

distribution function F(x), and let r* denote a uniformly distributed number drawn from

the unit interval. Then the probability of choosing x* as defined by

r *=F(x*)=

(6.3)

is f(x*). The theorem is graphically illustrated in figure 6.2 where the relationships among

the PDF, its associated CDF, the uniformly distributed number r* and the value x* of the

random variable X are shown.

Suppose that the function f(x) is a PDF and x [a,b]. The algorithm of the inversion

technique can be described in the next steps:

i. Check if f(x) is normalized. That is check if .

ii. Calculate the CDF: F(x) = .

iii. Generate random numbers R.

iv. Let F(x) =R and solve for x as x=F-1(R).

Figure 6.2. Graphical illustration of the relationships among the PDF, its associated

CDF, the uniformly distributed number r* and the value x* of the random variable X.

63

CDF 1

r*

x*

PDF

f(x*)

x*

x

x


6.4.2. The Rejection Method

The rejection method is an alternative when the inversion method cannot be implemented

(for example when the equation F(x)=R is difficult to solve). An important limitation

though of the rejection method is the fact that depending on the shape of the modeled PDF

it may be time consuming. If f(x) is a PDF and x [a, b] then the algorithm of the

rejection method is illustrated in figure 6.3 and it is described as the follows:

i. Find the maximum of f(x) and name it fo.

ii. Set g: g(x) = f(x)/fo.

iii. Generate a random number R1 [0,1).

iv. Generate a random value of x [a,b), say x*=a+(b-a)R1.

v. Generate a random number R2 [0,1).

vi. Compare R2 with g(x*):

If g(x*) R2 then: reject x* and return to step 3

else: accept x* and return to step 3.

Figure 6.3. Sampling a PDF f(x) by the rejection technique. The random pairs (x*, R2)

are assumed to be uniformly distributed over the circumscribing rectangle. Only those

bounded by f(x) are accepted.

64

(x*,R2)

b

f(x)/fo

Region of rejection

1

0 a x

Uniformly distributedordered pairs

Region of acceptance

CHAPTER 7: PRIMARY ELECTRON GENERATION MODEL

CHAPTER 7

PRIMARY ELECTRON

GENERATION MODEL

65


7.1. Introduction

This chapter deals with the Monte Carlo modeling of primary electron generation inside

the photoconductor’s bulk. Through experimental research is not feasible to isolate and

study only the primary electrons produced inside a photoconductor. Thus, a complete

validation of the model cannot be done. Nevertheless, an indirect index of the reliability of

the method rises from the fact that the developed model is an extension of a recently

presented model to simulate the primary electron production inside a-Se (Sakellaris et al

2005), which is based on a validated model developed by Spyrou et al (1998) that

simulates the x-ray energy spectrum sampling as well as the x-ray photon interactions.

This section focuses on the simulation of primary electron production from x-ray-matter

interactions (incoherent scattering, photoelectric absorption) as well as due to atomic

deexcitation (fluorescent photon production, Auger and CK electron emission) inside

a-Se, a-As2Se3, GaSe, GaAs, Ge, CdTe, CdZnTe, Cd0.8Zn0.2Te, ZnTe, PbO, TlBr, PbI2 and

HgI2.

Figure 7.1 presents the flowchart of interaction processes taken into account in the

simulation. The photon interaction cross sections for photoelectric absorption, coherent

and incoherent scattering are extracted from XCOM (Berger et al 2005), the atomic form

factors and incoherent scattering functions from Hubbel et al (1975), while the shell and

subshell binding energies from Bearden and Burr (1967).

7.2. Electron from Incoherent Scattering

The energy of the recoil electron (Ee) is given by the following equation:

(7.1)

where Ep is the energy of the initial photon, mc2 is the electron rest energy and θp is the

polar angle of the scattered photon. Thus, using the sampled values of Ep, θp (Spyrou et al

1998), random samples of Ee are taken.

The direction of the recoil electron is calculated by sampling both the azimuthal φ e

and the polar angle θe of the electron. The azimuthal angle φe is uniformly distributed in

the interval [0,2π). Thus, the inversion method is used to sample the azimuthal PDF given

by:

66

PDF(φe)= (7.2)


(7.2)

Figure 7.1. A flowchart of the interaction processes taken into account in the simulation

model.

Random samples of polar angle θe are taken from equation below:

(7.3)

which is derived from angle calculations in the scattering process.

7.3. Photoelectric absorption

7.3.1. Photoelectric absorption from a molecule

For the case of compound materials, the molecular photoelectric cross section is evaluated

as the weighted sum of the photoelectric cross sections of the atomic constituents

(additivity approximation). Therefore if AxBy is the compound, then the molecular

photoelectric cross section of AxBy at energy E, , is defined as:

(7.4)

67

Primary Photon

Coherent Scattering

Scattered photon

Incoherent Scattering

Scattered photon

Ejected electron

Photoelectric Absorption

Emitted photoelectron

Atomic Deexcitation

Fluorescent photon

Auger electron

Coster-Kronig electron

Electron Interactions


where are the photoelectric cross sections (in cm2/g) and wA, wB the

fractions by weight of elements A and B, respectively. If the probability of a photon to

interact with atom A is defined as:

(7.5)

then . Thus, the atom which photoelectrically absorbs the photon is

determined by a Monte Carlo decision, based on the probabilities and .

7.3.2. Production of photoelectron

Similar to the case of a-Se (Sakellaris et al 2005), it has been assumed that photons with

energies hν 1.434 keV, which is the binding energy of Se LIII subshell, are not taken into

account in the simulation process.

For the case of Se the differences between the binding energies of L subshells as

well as the binding energies of M subshells are smaller or equal to 1 keV and were

assumed to be negligible (Sakellaris et al 2005). Since this is also the case in the rest of

the elements except for the heavy ones (Hg, Tl, Pb), in order to determine the shell

(or subshell) from which the photoelectron is ejected, the formulation followed for a-Se

(Sakellaris et al 2005) can be adopted. Therefore:

i. If hν > BK, where BK is the binding energy of the K shell, the photon is absorbed by

the K shell, ejecting a photoelectron with energy Ee= hν-BK.

ii. If BLIII<h BK, the photon is absorbed by the LIII subshell (representing the L shell),

ejecting a photoelectron with energy Ee= hν-BLIII.

iii. If 1.434 keV<hν BLIII, as it can be the case for Br, Cd, Te and I that have L III

binding energies 1.550 keV, 3.538 keV, 4.341 keV and 4.557 keV respectively, the

photon is absorbed by an outer shell (M, N), ejecting a photoelectron with energy

Ee=hν.

On the other hand in the case of the heavy elements (Hg, Tl, Pb), the differences

between the binding energies of L subshells as well as the binding energies of M subshells

are larger than 1 keV and therefore the above formulation cannot be used. In these

elements, the shell that absorbs the photon is determined by Monte Carlo sampling of the

subshells photoelectric cross sections extracted from the Evaluated Photon Data Library

EPDL97 of the Lawrence Livermore National Laboratory (Cullen et al 1997). In

68


particular, if the total photoelectric cross section at energy E is and the subshell ‘s’

photoelectric cross section is then:

(7.6)

If the subshell photoelectric probability is defined as:

(7.7)

then . Thus, the subshell ‘s’ that photoelectrically absorbs the photon is

determined by a Monte Carlo decision, based on the probabilities . After the subshell

‘s’ selection, a photoelectron is ejected from that subshell with energy Ee=hν-Bs for L and

M shells, and with energy Ee=hν for N and O outer shells. Since the photon energies

considered in the simulation are lower than the binding energies of the K shells of Hg, Tl

and Pb (83.102 keV, 85.530 keV and 88.005 keV respectively), these shells do not

contribute in the photoelectric effect.

The direction of the photoelectron is described by the polar angle θ and the

azimuthal angle φ, in a coordinate system that has its origin at the interaction point and its

z-axis along the initial photon direction. Thus, the direction is sampled, using Monte Carlo

techniques, from the following equation (Davisson and Evans 1952):

(7.8)

where dI/dΩ is the number of photoelectrons per solid angle, A is a constant and β=υ/c

where υ is the electron velocity.

One of the ways to sample both θ and φ from this equation is by using the rejection

method. The differential cross section for an atom can be considered to be the product of

two independent PDFs:

i. The azimuthal PDF normalized to a maximum of unity:

h1(φ) = cos2φ (7.9)

ii. The polar PDF normalized to a maximum of unity:

(7.10)

where h(θ) is given as:

69


(7.11)

and hmax(θ) is the maximum value of h(θ), calculated by taking its first derivative equal to

zero.

7.4. Atomic deexcitation

After having interacted with photoelectric effect, the atom deexcites through radiative and

non-radiative transitions in which the vacancies produced migrate to outer shells. The

radiative transitions are the emission of fluorescent photons and the non-radiative

transitions are the emission of Auger and CK electrons.

In the simulation process, only the deexcitation of K and L shells has been

considered. In particular, as it was the case for a-Se (Sakellaris et al 2005) the deexcitation

of L shell in Ga, As, Ge and Zn has been disregarded due to the fact that the energy

released is lower than 1.434 keV. For the rest of the elements the L shell deexcitation has

been taken into account. Therefore, the atomic deexcitation cascade is simulated until the

vacancies have migrated to M and outer shells or until the deexcitation energy has fallen

down the considered threshold of 1.434 keV. The types of atomic transitions and their

probabilities (fluorescence, Auger and Coster-Kronig yields) are taken from the Evaluated

Atomic Data Library (EADL) of the Lawrence Livermore National Laboratory

(Perkins et al 1991, Cullen 1992).

7.4.1. K and L shell deexcitation

The simulation of K shell deexcitation (fluorescence or Auger electron emission) for all

the elements is the same as for the case of a-Se (Sakellaris et al 2005). Therefore, the K

shell releases its excitation energy as follows:

i. Emission of a K-fluorescence photon, with probability PF=FKωK, where FK is the

fraction of the photoelectric cross section contributed by K-shell electrons (obtained

from Storm and Israel (1970)) and ωK is the K-fluorescent yield.

ii. Emission of an Auger electron, with probability PA=1- PF.

Since these two phenomena are complementary (PF+PA=1), the type of atom’s secondary

interaction is determined by a Monte Carlo decision, based on the probabilities PF and

PA. Table 7.1 gives the values of FK and ωK for the various elements (except for Hg, Tl

and Pb in which the K shell does not take part in the photoelectric absorption).

70


Table 7.1. The fraction of the photoelectric cross section contributed by K-shell electrons

(FK) and the K-fluorescent yield (ωK) for the various elements (except for Hg, Tl and Pb).

The type of L shell’s deexcitation mechanism (fluorescence, Auger and CK electron

emission) is determined by a Monte Carlo decision based on the fluorescence, Auger and

CK yields. Table 7.2 gives the values of fluorescent (ω), Auger (α) and CK (ck) yields of

L subshells for Br, Cd, Te, I, Hg, Tl and Pb, in which the L shell deexcitation has been

considered.

Table 7.2. The fluorescence (ωLI, ωLII, ωLIII), the Auger (aLI, aLII, aLIII) and the Coster-

Kronig yields (ckLI, ckLII) of subshells LI, LII and LIII for Br, Cd, Te, I, Hg, Tl and Pb.

7.4.2. Simulated atomic transitions

When the atom’s deexcitation mechanism is determined, a decision on the particular

atomic transition that occurs is made. Since the fluorescence yield ωs, the Auger yield as

and CK yield cks of a shell (or subshell) ‘s’ are the sum of the yields of all possible

fluorescent, Auger and CK transitions ‘j’ pertinent to that shell, the atomic transition that

71

Element FK ωK

Zn 0.870 0.466Ga 0.869 0.497Ge 0.867 0.528As 0.866 0.557Se 0.864 0.596Br 0.864 0.613Cd 0.841 0.843Te 0.836 0.879I 0.835 0.886

ElementFluorescence yield Auger yield Coster-Kronig yield

ωLI ωLII ωLIII aLI aLII aLIII ckLI ckLII

Br 0.003 0.019 0.019 0.185 0.896 0.981 0.812 0.085Cd 0.021 0.059 0.061 0.298 0.778 0.939 0.681 0.163Te 0.040 0.079 0.081 0.438 0.747 0.919 0.522 0.174I 0.043 0.085 0.086 0.432 0.740 0.914 0.525 0.175Hg 0.086 0.376 0.330 0.146 0.497 0.670 0.767 0.126Tl 0.091 0.390 0.341 0.147 0.486 0.659 0.762 0.124Pb 0.098 0.404 0.352 0.148 0.475 0.648 0.753 0.122


occurs is determined by a Monte Carlo decision based on the probabilities , where

y {ω, a, ck} and , requiring .

The atomic transitions that have been taken into account for a particular deexcitation

mechanism and shell (or subshell) in the simulation process were selected according to

their probability of occurrence and their energy. The energy resolution considered in our

statistics was 1 keV. Transitions that their energy lies in the same energy range, for

example in 12-13 keV, were grouped together and the most probable among them was

chosen to be the representative one. An example is given in table 7.3 for the case of I.

Table 7.3. The probabilities for fluorescence, Auger and CK electron emission forK and L shells and the simulated atomic transitions with the corresponding energies and

probabilities for the case of I.

72

Probability ofAtomic Deexcitation

Simulated Atomic Transition

Transition's Probability

Transition's Energy (keV)

FKωΚ 0.740 KLIII 0.820 28.612KMIII 0.147 32.294KNIII 0.033 33.046

1- FKωΚ 0.260 KLILI 0.078 22.793KLIILIII 0.466 23.760KLIIILIII 0.124 24.055KLIMI 0.028 26.909KLIIIMIII 0.234 27.737KLIIINIII 0.037 28.489KMIIMIII 0.024 31.363KMIIINIII 0.008 32.171

ωLI 0.043 LILIII 0.009 0.631LIMIII 0.806 4.313LINIII 0.186 5.065

αLI 0.432 LIMIVMV 0.797 3.938LIMIVNV 0.198 4.508LINIVNV 0.005 5.089

ckLI 0.525 LILIIINV 1.000 0.582ωLII 0.085 LIIMI 0.047 3.780

LIIMIV 0.953 4.221αLII 0.740 LIIMIMII 0.038 2.849

LIIMIVMV 0.819 3.602LIIMIVNV 0.142 4.172

ckLII 0.175 LIILIIINIV 1.000 0.244ωLIII 0.086 LIIIMV 0.880 3.938

LIIINV 0.120 4.508αLIII 0.914 LIIIMIVMV 0.781 3.307

LIIINIVNV 0.016 4.458LIIIMIIIMIII 0.203 4.557


7.4.3. Energies and directions of fluorescent photons, Auger and CK electrons

As it was mentioned in chapter 5, the energy of emitted fluorescent photons, Auger and

CK electrons is the difference between the binding energies of the shells that are

involved in the particular transition. Therefore, if the emission of a fluorescent photon

involves shells i and j then the energy of the fluorescent photon (Efl;ph) is given by Efl;ph=Bi-

Bj, whereas if shells i, j and k take part in the emission of an Auger or CK electron, then

the energy of Auger or CK electron (EAug/CK) is given by EAug/CK=Bi-Bj-Bk.

Both fluorescent photons and Auger and CK electrons are isotropically ejected from

the atom. The normalized PDF is given by:

(7.12)

and it can be considered as the product of two independent normalized PDFs

(PDF(θ)=1/2sinθ and PDF(φ)=1/2π) which are sampled, using the inversion method, to

produce the proper values of θ and φ.

7.5. Model limitations

The model presented is based on assumptions arising from the compromise between

accuracy and algorithmic simplicity. In particular, for all the elements except for Hg, Tl

and Pb, it has been assumed that the photoelectric absorption of a photon with energy

higher than the binding energy of K shell occurs with that shell only. It has been

calculated that for a-Se at 20 keV (average mammographic energy) the absolute value of

the relative difference (NK-NS)/NS between the number of primary electrons produced

using the above assumption (NK) and that number using subshell photoelectric cross

sections (NS) is 2.128 %. Additionally, the absolute value of the relative difference in the

total energy of primary electrons was calculated to be 2.33 %.

Furthermore, the deexcitation of M and outer shells has not been taken into account.

In all the elements except for Hg, Tl and Pb, the deexcitation energy released from these

shells is lower than 1.434 keV. In Hg, Tl and Pb though, the M shells can deexcite by

releasing energy which is higher than 1.434 keV since the M subshells have binding

energies between 2 and 4 keV. This means that there is an underestimation in the number

of primary electrons with energies E< 4 keV for Hg, Tl and Pb especially for incident

x-ray energies which are lower than the LIII subshell binding energies of these elements

(13.035 keV for Pb, 12.284 keV for Hg and 12.658 keV for Tl). Nevertheless, since the

average mammographic energy is of the order of 20 keV whereas the low energy photons

73


are strongly absorbed in the breast, this underestimation is not considered to be important

compared to the attempt to keep the algorithmic complexity in feasible levels.

74


CHAPTER 8

PRIMARY ELECTRON

GENERATION:

RESULTS & DISCUSSION

75


8.1. Introduction

Based on the method described in Chapter 7, a Monte Carlo code has been developed in

order to run a set of in silico experiments using 39 monoenergetic spectra, with energies

between 2 and 40 keV, and 53 mammographic spectra, in which the majority of photons

have energies between 15 and 40 keV obtained from Fewell and Shuping (1978). The list

of mammographic spectra is presented in table 8.1.

Table 8.1. The 53 mammographic spectra used in the simulation process.

The x-ray photons (107 in number) are incident at the center of a detector with

dimensions 10 cm width, 10 cm length and 1 mm thickness, consisting of the already

mentioned set of materials. The choice of 1 mm thickness of the photoconductors was

76

Spectral Group (Anode Material/

1st Filter/ 2nd Filter)

Spectra (Anode Material/


Spectral Group (Anode Material/


Spectra (Anode Material/


20 kVp Mo/ 0/ 0 25 kVp W/ 0.51mm Al/ 0 25 kVp Mo/ 0/ 0 25 kVp W/ 1.02mm Al/ 0 30 kVp Mo/ 0/ 0 25 kVp W/ 3.05mm Al/ 0 35 kVp Mo/ 0/ 0 30 kVp W/ 0.51mm Al/ 0

Mo/ 0/ 0

40 kVp Mo/ 0/ 0 30 kVp W/ 1.02mm Al/ 0 25kVp Mo/ 0.51mm Al/ 0 30 kVp W/ 2.03mm Al/ 0 30kVp Mo/ 0.51mm Al/ 0 30 kVp W/ 3.05mm Al/ 0 30kVp Mo/ 1.02mm Al/ 0 35 kVp W/ 1.02mm Al/ 0 35kVp Mo/ 0.51mm Al/ 0 40 kVp W/ 0.51mm Al/ 0 40kVp Mo/ 0.51mm Al/ 0 40 kVp W/ 1.02mm Al/ 0

Mo/ Al/ 0

40kVp Mo/ 1.52mm Al/ 0 40 kVp W/ 2.03mm Al/ 0 30kVp Mo/ 0.51mm Al/ 0.030mm Mo

W/ Al/ 0

40 kVp W/ 3.05mm Al/ 0 35kVp Mo/ 0.51mm Al/ 0.030mm Mo 30 kVp W/ 0/ BS 0.13mm of SN 40kVp Mo/ 0.51mm Al/ 0.030mm Mo 40 kVp W/ 0/ BS 0.13mm of SN

Mo/ Al/ Mo

40kVp Mo/ 1.02mm Al/ 0.030mm Mo W/ 0/ BS*

40 kVp W/ 0/ BS 0.13mm of LA 20kVp Mo/ 0/ 0.030mm Mo 25kVp Mo/ 0/ 0.030mm Mo

25kVp MoW/ 0.51mm Al/ 0

25kVp Mo/ 0/ 0.015mm Mo 30kVp MoW/ 0.51mm Al/ 0 30kVp Mo/ 0/ 0.015mm Mo 30kVp MoW/ 1.02mm Al/ 0 30kVp Mo/ 0/ 0.030mm Mo 35kVp MoW/ 0.51mm Al/ 0 35kVp Mo/ 0/ 0.030mm Mo 40kVp MoW/ 0.51mm Al/ 0 40kVp Mo/ 0/ 0.015mm Mo 40kVp MoW/ 1.02mm Al/ 0

Mo/ 0/ Mo

40kVp Mo/ 0/ 0.030mm Mo 40kVp MoW/ 2.03mm Al/ 0 25 kVp W/ 0/ 0

MoW/ Al/ 0

40kVp MoW/ 1.52mm Al/ 0 30 kVp W/ 0/ 0 30kVp MoW/ 0.51mm Al/ 0.030mm Mo 35 kVp W/ 0/ 0

MoW/ Al/ Mo 40kVp MoW/ 0.51mm Al/ 0.030mm Mo

W/ 0/ 0

40 kVp W/ 0/ 0 MoW/ 0/ Mo 30kVp MoW/ 0/ 0.030 mm Mo *Beam Shaping


made so that the number of both primary and fluorescent photons that escape forwards to

be negligible, since primary and fluorescent photons are the major sources of primary

electron production from their photoelectric absorption.

The results obtained are grouped in four categories:

A. Energy distributions of: (i) fluorescent photons, (ii) primary and fluorescent

photons escaping forwards and backwards, (iii) primary electrons.

B. Azimuthal and polar angle distributions of primary electrons.

C. Spatial distributions of primary electrons.

D. Arithmetics of: (i) fluorescent photons, (ii) primary and fluorescent photons

escaping forwards and backwards, (iii) primary electrons.

8.2. Energy distributions.

8.2.1. Fluorescent photons.

The distributions for CdZnTe and Cd0.8Zn0.2Te are similar. Thus, in figure 8.1 the energy

distributions of fluorescent photons at 40 keV incident x-ray energy are shown for all the

materials except for Cd0.8Zn0.2Te. The bin size used in the energy distribution histograms

was 1 keV. The incident energy of 40 keV has been chosen in order to let all the possible

fluorescent transitions to occur and thus the corresponding spectral peaks to be presented.

8.2.2. Escaping photons.

In all materials and incident x-ray spectra, fluorescent photons escape backwards. The

energy distributions of primary photons that escape backwards resemble the shape of the

incident spectrum, while this is not the case for primary photons that escape forwards. The

forwards escaping primary photons have relatively high energies and well above the

absorption edges.

As characteristic examples the energy distributions of primary and fluorescent

photons that escape forwards and backwards in CdTe for an incident spectrum resulting

from Mo, at 40 kVp, with half value layer (HVL): 0.68 mm Al and filter Al: 0.51 mm, are

presented in figures 8.2 and 8.3, respectively.

8.2.3. Primary electrons.

Since the photoelectric absorption is the dominant interaction mechanism between x-rays

and matter in the mammographic energy range, the primary electrons are consisted of

77


photoelectrons, Auger and CK electrons. The energy distributions of primary electrons

are characterised from the presence of certain spectral peaks which are due to

Figure 8.1. The energy distributions of fluorescent photons at 40 keV for (a) a-Se,

(b) a-As2Se3, (c) GaSe, (d) GaAs, (e) Ge, (f) CdTe, (g) CdZnTe, (h) ZnTe, (i) PbO, (j) TlBr,

(k) PbI2 and (l) HgI2.

the atomic deexcitations of the material being irradiated. Therefore these peaks are

due to the photoelectrons produced by the absorption of fluorescent photons as well as

due to the Auger and CK electrons. The distributions are also filled in with photoelectrons

produced by the absorption of primary photons. Thus, their energies depend on the

78

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41Energy (keV)

Num

ber o

f flu

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cent

pho

tons

x 10

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0

0.5

1

1.5

2

2.5

3

3.5

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41Energy (keV)

Num

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f flu

ores

cent

pho

tons

x 10

60

0.5

1

1.5

2

2.5

3

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41Energy (keV)

Num

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f flu

ores

cent

pho

tons

x 10

6

0

0.5

1

1.5

2

2.5

3

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41Energy (keV)

Num

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cent

pho

tons

x 10

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0

0.5

1

1.5

2

2.5

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1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41Energy (keV)

Num

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f flu

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x 10

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0

0.5

1

1.5

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3

3.5

4

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41Energy (keV)

Num

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x 10

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0

0.5

1

1.5

2

2.5

3

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41Energy (keV)

Num

ber o

f flu

ores

cent

pho

tons

x 10

6

00.5

11.5

22.5

33.5

44.5

55.5

6

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41Energy (keV)

Num

ber o

f flu

ores

cent

pho

tons

x 10

6

0

0.5

1

1.5

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41Energy (keV)

Num

ber o

f flu

ores

cent

pho

tons

x 10

6

0

0.5

1

1.5

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41Energy (keV)

Num

ber o

f flu

ores

cent

pho

tons

x 10

6

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41Energy (keV)

Num

ber o

f flu

ores

cent

pho

tons

x 10

6

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41Energy (keV)

Num

ber o

f flu

ores

cent

pho

tons

x 10

6

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)


incident spectrum. The mean fraction of incident x-ray energy transferred to primary

electrons is 97% whereas the minimum is 84.5% (CdTe at 32 keV).

Figure 8.2. (a) An incident x-ray spectrum resulting from Mo, kVp: 40, HVL: 0.68 mm Al,

filter Al: 0.51 mm and (b) the corresponding energy distributions of primary photons that

escape forwards and backwards in CdTe.

Figure 8.3. The energy distributions of fluorescent photons that escape forwards and

backwards in CdTe for an incident x-ray spectrum resulting from Mo, kVp: 40,

HVL: 0.68 mm Al, filter Al: 0.51 mm.

8.2.3.1. Monoenergetic case.

The distributions for CdZnTe and Cd0.8Zn0.2Te are similar. Thus, in figure 8.4 the energy

distributions of primary electrons at 40 keV incident x-ray energy are shown for all

the materials except for Cd0.8Zn0.2Te. The peaks which are due to the atomic deexcitations

are shown in black color whereas the peaks that correspond to photoelectrons from the

primary photon absorption are shown in white color. The incident energy of 40 keV has

79

020406080

100120140160180200

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41Energy (keV)

Num

ber

of fl

uore

scen

t pho

tons

esc

apin

g x

103

ForwardsBackwards

0

1

2

3

4

5

6

7

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41Energy (keV)

Num

ber o

f pri

mar

y ph

oton

s esc

apin

g x

103

ForwardsBackwards Cd K-edge

Te K-edge

0

0.5

1

1.5

2

2.5

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 3133 35 3739 41Energy (keV)

Num

ber

of p

rimar

y ph

oton

s x 1

06

(a) (b)


been chosen in order to let all the possible atomic transitions to occur and thus the

corresponding spectral peaks to be presented.

Figure 8.4. The energy distributions of primary electrons at 40 keV for (a) a-Se,

(b) a-As2Se3, (c) GaSe, (d) GaAs, (e) Ge, (f) CdTe, (g) CdZnTe, (h) ZnTe, (i) PbO, (j) TlBr,

(k) PbI2 and (l)HgI2. The peaks in black color are due to the atomic deexcitations whereas

the peaks in white color correspond to photoelectrons from the absorption of primary

photons.

8.2.3.2. Polyenergetic case.

The shape of the energy distributions for the case of a-Se, a-As2Se3, GaSe, GaAs and Ge

for all the polyenergetic x-ray spectra resembles the shape of the incident spectrum shifted

80

0

1

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3

4

5

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7

8

1 3 5 7 9 11 1315 17 1921 2325 2729 3133 3537 39 41Energy (keV)

Num

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ary

electr

ons x

106

0

0.51

1.52

2.53

3.54

4.55

5.56

1 3 5 7 9 11 13151719212325272931 33353739 41Energy (keV)

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elect

rons

x 10

6

00.5

11.5

22.5

33.5

44.5

55.5

6

1 3 5 7 9 11 131517192123252729313335373941Energy (keV)

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f prim

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elect

rons

x 10

6

0123456789

10

1 3 5 7 9 11 1315 17192123252729 3133353739 41Energy (keV)

Num

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f prim

ary

electr

ons x

106

0123456789

101112

1 3 5 7 9 11 131517 19 212325272931333537 3941Energy (keV)

Num

ber o

f prim

ary e

lectr

ons x

10

6

0123456789

10

1 3 5 7 9 11 13151719 2123252729 3133353739 41Energy (keV)

Num

ber o

f prim

ary

elect

rons

x 10

6

0123456789

10

1 3 5 7 9 11 13151719212325272931 3335 373941Energy (keV)

Num

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f prim

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electr

ons x

106

0

0.51

1.52

2.53

3.54

4.55

1 3 5 7 9 11 13151719 212325 272931 33353739 41Energy (keV)

Num

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f prim

ary

elect

rons

x 10

6

0

0.5

1

1.5

2

2.5

3

3.5

1 3 5 7 9 11131517192123 252729313335 373941Energy (keV)

Num

ber o

f prim

ary

elect

rons

x 1

06

0

1

2

3

4

5

6

7

8

1 3 5 7 9 11 131517 19212325272931 33353739 41Energy (keV)

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106

0

1

2

3

4

5

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7

8

1 3 5 7 9 11 13151719 212325 272931 33353739 41Energy (keV)

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6

0123456789

10

1 3 5 7 9 11 1315 17 19 21 23 25 27 29 3133 35 37 39 41Energy (keV)

Num

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elect

rons

x 10

6

(a)

(d)

(g)

(b)

(e)

(h)

(c)

(f)

(i)

(j) (l) (k)

of photons have energies lower than the binding energies of Cd and Te K shells

(26.711 keV and 31.814 keV respectively).


at lower energies with the peaks due to the atomic deexcitations added. This is also the

case for CdTe, CdZnTe, Cd0.8Zn0.2Te and ZnTe for incident spectra in which the majority

Figure 8.5. The energy distributions of primary electrons in (a) CdZnTe for an x-ray

spectrum resulting from W, kVp: 30, HVL: 0.81 mm Al, filter Al: 1.02 mm, (b) CdZnTe for

an x-ray spectrum resulting from W, kVp: 40, HVL: 1.22 mm Al, filter Al: 1.02 mm,

(c) PbI2 for an x-ray spectrum resulting from Mo, kVp: 30, HVL: 0.5 mm Al, filter

Mo: 0.03 mm, (d) PbI2 and (e) PbO for an x-ray spectrum resulting from W, kVp: 40, HVL:

1.22 mm Al, filter Al: 1.02 mm.

A representative result is shown in figure 8.5(a) that presents the energy distribution

of primary electrons in CdZnTe for an incident spectrum resulting from W, at 30 kVp with

81

0

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1.5

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2.5

3

3.5

4

4.5

5

5.5

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Num

ber

of p

aric

les x

10

6

Primary photonsPrimary electrons

0

1

2

3

4

5

6

7

8

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41Energy (keV)

Num

ber

of p

aric

les x

10

6


0

0.5

1

1.5

2

2.5

3

3.5

4

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41Energy (keV)

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of p

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10

6


0

0.5

1

1.5

2

2.5

3

3.5

4

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41Energy (keV)

Num

ber

of p

aric

les x

10

6Primary photonsPrimary electrons

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41Energy (keV)

Num

ber

of p

aric

les x

10

6


(a) (b)

(c) (d)

(e)


half value layer (HVL): 0.81 mm Al and filter Al: 1.02 mm. As the number of incident

photons with energies higher than Cd and Te K edges increases, the resemblance between

the incident spectrum and the resulting electron distribution decreases and the deexcitation

peaks become the characteristic feature. A representative result is shown in figure 8.5(b)

that presents the energy distribution of primary electrons in CdZnTe for another incident

spectrum resulting from W, kVp: 40, HVL: 1.22 mm Al, filter Al: 1.02 mm.

For the case of PbI2 and HgI2 the atomic deexcitation peaks are the characteristic

feature of the energy distributions for all the incident polyenergetic spectra. As

representative results the energy distribution of primary electrons in PbI2 is shown in

figure 8.5(c) for an incident spectrum resulting from Mo, kVp: 30, HVL: 0.5 mm Al, filter

Mo: 0.03 mm and in figure 8.5(d) for an incident spectrum resulting from W, kV p: 40,

HVL: 1.22 mm Al, filter Al: 1.02 mm. This is also the case for PbO and TlBr for all the

incident polyenergetic spectra except for those in which the majority of photons have

energies higher than 25 keV. For these spectra photoelectrons can also be produced at

energies where no deexcitation peaks are present. Therefore this case is similar to the case

of a-Se, a-As2Se3, GaSe, GaAs and Ge described above. A representative result is

shown in figure 8.5(e) that presents the energy distribution of primary electrons in PbO for

an incident spectrum resulting from W, kVp: 40, HVL: 1.22 mm Al, filter Al: 1.02 mm.

The mammographic spectra chosen to describe the spectral dependence of primary

electron energy distributions in figure 8.5 were selected for this dependence to be clearly

illustrated and to allow a comparison between the various photoconductors.

8.3. Angular distributions of primary electrons.

In order to study the directions of the primary electrons produced, two histograms were

plotted: that of azimuthal angle φ and that of the cosine of the polar angle θ. The angles

were calculated in a spherical coordinate system having z-axis perpendicular to the

detector plane and direction that of incident photons.

8.3.1. Azimuthal distributions.

In figure 8.6 the electron azimuthal distributions in CdZnTe for all the monoenergetic

spectra are shown. The figure is a representative result for the shape and energy

dependence of azimuthal distributions. The shape of azimuthal distributions corresponds

to the plot of the azimuthal probability density function (PDF) for the photoelectric

82


process, PDF(φ) = (Sakellaris et al 2005), an expected fact since in the

mammographic energy range the photoelectric effect dominates. Thus, the primary

Figure 8.6. The azimuthal distributions of primary electrons in CdZnTe for incident

monoenergetic spectra with energies between 2 and 40 keV. The distributions form groups

which separate at 4, 5, 10, 27 and 32 keV.

electrons have the maximum probability to be ejected at φ=0, π and 2π and the minimum

one at φ=π/2 and 3π/2.

In figure 8.6 it is seen that at energies where there is no atomic deexcitation, as it is

the case for CdZnTe at E 3 keV, the minima of the distributions are close to zero. On

the other hand, at energies where atomic deexcitation is present, for example at E 4 keV

in the case of CdZnTe, a background is added. This background is due to the emitted

Auger and CK electrons, which are ejected isotropically (Sakellaris et al 2005) and hence

are uniformly distributed over the azimuthal angles. The higher is the number of Auger

and CK electrons produced, the wider the background added and thus the higher the

distributions shift, forming separated groups. For example in the case of CdZnTe the

distributions form six groups separated at 4, 5, 10, 27 and 32 keV where which Cd and Te

LIII subshells, Zn, Cd and Te K shells are excited, respectively. The groups lie within

certain zones. The larger the differences in the number of electrons among the

distributions in the same group the wider the zone. When the number of electrons

increases as the incident energy increases, the distributions in a certain group shift

83

05

10152025303540455055606570758085

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6Azimuthal angle φ (rad)

Num

ber

of p

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s x

103

2 keV3 keV4 keV5 keV6 keV7 keV8 keV9 keV10 keV11 keV12 keV13 keV14 keV15 keV16 keV17 keV18 keV19 keV20 keV21 keV22 keV23 keV24 keV25 keV26 keV27 keV28 keV29 keV30 keV31 keV32 keV33 keV34 keV35 keV36 keV37 keV38 keV39 keV40 keV

4 keV5 keV10 keV

27 keV

32 keV


upwards. For example this is the case in CdZnTe for the three groups with E 10 keV.

Similarly, when the number of electrons decreases as the energy increases the distributions

Figure 8.7. (a) The azimuthal distributions of primary electrons for a-Se, HgI2 and CdTe

at 30 keV normalized at their maxima. (b) The histogram of the minima of the normalized

azimuthal distributions for the various materials at 30 keV.

shift downwards. The zones become wider the closer the azimuthal angles are to 0, π and

2π, because at these angles the photoelectrons have increased probability to be ejected.

Due to the fact that the Auger and CK electrons are uniformly ejected (with the same

probability) at the various azimuthal angles, the larger their contribution is in the number

of primary electrons the higher the azimuthal uniformity in electron directions. That is,

when their contribution increases the probabilities of electron ejection at the various

azimuthal angles increase, especially the closer these angles are to π/2 and 3π/2, and tend

to become equal to the probability of ejection at 0, π and 2π. In other words, higher

azimuthal uniformity means smaller tendency of electron ejection at 0, π and 2π. These

can be seen in figure 8.7 that presents the azimuthal distributions for a-Se, HgI2 and CdTe

normalized at their maxima (figure 8.7(a)) and the histogram of the minima of the

normalized distributions for the various materials (figure 8.7(b)), at 30 keV incident x-ray

energy. At this energy, the azimuthal uniformity in CdZnTe, Cd0.8Zn0.2Te and CdTe is

higher compared to the rest of materials because Cd K and L shells deexcite emitting a

large number of Auger and CK electrons. Therefore the minima of the corresponding

normalized azimuthal distributions are closer to unity. For similar reasons at E 32 keV

the azimuthal uniformity in ZnTe, PbI2 and HgI2 significantly increases. It was found that

for the practical mammographic energies (15 keV E 40 keV), and therefore for all the

84

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

a-Se a-As2Se3 Ge PbO GaSe GaAs PbI2 HgI2 TlBr ZnTe CdZnTe Cd08Zn02Te CdTe

Material

Min

ima

of n

orm

alized

azi

mut

hal

dist

ribu

tions

(b)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6Azimuthal angle φ (rad)

Nor

mal

ized

num

ber

of p

rim

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s

a-Se

HgI2

CdTe

(a)


polyenergetic spectra, a-Se, a-As2Se3 and Ge have the minimum azimuthal uniformity

whereas CdZnTe, Cd0.8Zn0.2Te and CdTe the maximum one.

Figure 8.8. The polar distributions (cosθ) of primary electrons for GaAs at

(a) 2 keV, (b) 5 keV, (c) 8 keV and (d) 10 keV. In GaAs at E 10 keV there is not atomic

deexcitation and therefore the distributions are influenced only by the photoelectric effect.

The azimuthal uniformity together with the rest of the analysis made for azimuthal

angles and the analysis that will be made in next section for the polar angles defines, in the

presence of an electric field, the trajectories of primary electrons in the bulk and

consequently is one of the factors that affect the final image characteristics.

8.3.2. Polar distributions.

The monoenergetic case reveals the fact that the polar distributions are affected by two

factors: the photoelectric effect and the atomic deexcitation. As a characteristic example

the case of GaAs (figures 8.8 and 8.9) is discussed.

85

0

1

2

3

4

5

6

7

8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1cosθ

Num

ber

of p

rim

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elec

tron

s x 1

03

Most probable cosθ

0

1

2

3

4

5

6

7

8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1cosθ

Num

ber

of p

rim

ary

elec

tron

s x 1

03

Most probable cosθ

0

1

2

3

4

5

6

7

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1cosθ

Num

ber

of p

rim

ary

elec

tron

s x 1

03

Most probable cosθ

0

1

2

3

4

5

6

7

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1cosθ

Num

ber

of p

rim

ary

elec

tron

s x 1

03

Most probable cosθ

(a) (b)

(c) (d)


The distributions have minima at cosθ=-1 and 1 that is the primary electrons have

the minimum probability to be ejected parallel to the incident beam’s axis either forwards

or backwards. Furthermore, the number of electrons in the positive cosine values is higher

than the number in the negative values which means that the electrons prefer to be emitted

Figure 8.9. The polar distributions (cosθ) of primary electrons for GaAs at (a) 12 keV,

(b) 20 keV, (c) 30 keV and (d) 40 keV. In GaAs at E 12 keV the distributions are

additionally influenced by the atomic deexcitation.

forwards. Actually, they prefer to be emitted at a particular polar angle θ that corresponds

to the most probable cosine value indicated in the figures.

As the energy of the photoelectrons increases, the distributions shift further to the

positive cosine values, that is the probability for forward ejection increases. Furthermore,

the most probable cosθ increases which means that the corresponding polar angle

decreases. At energies where the atomic deexcitation is present (figure 8.9) a background

is added. Hence, the probability of primary electrons to be ejected at parallel directions to

the incident beam’s axis increases. Since the fluorescent photons as well as the Auger and

86

0

2

4

6

8

10

12

14

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1cosθ

Num

ber

of p

rim

ary

elec

tron

s x 1

03

Most probable cosθ

0

2

4

6

8

10

12

14

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1cosθ

Num

ber

of p

rim

ary

elec

tron

s x 1

03

Most probable cosθ

0

2

4

6

8

10

12

14

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1cosθ

Num

ber

of p

rim

ary

elec

tron

s x 1

03

Most probable cosθ

0

2

4

6

8

10

12

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1cosθ

Num

ber

of p

rim

ary

elec

tron

s x 1

03

Most probable cosθ

(a) (b)

(c) (d)


CK electrons are isotropically ejected (the normalized polar PDF=sinθ= ), this

background is due to the Auger and CK electrons emitted at the point of x-ray incidence

as well as due to the primary electrons produced by the absorption of fluorescent photons.

Figure 8.10. The polar distribution (cosθ) of primary electrons in HgI2 for an x-ray

spectrum resulting from Mo, kVp: 30, HVL: 0.5 mm Al, filter Mo: 0.03 mm. The

distribution is a characteristic example for the polyenergetic case.

Figure 8.10 presents the polar distribution of primary electrons in HgI2 for an x-ray

spectrum resulting from Mo, kVp: 30, HVL: 0.5 mm Al, filter Mo: 0.03 mm and is a

characteristic example for the polyenergetic case. In agreement to the analysis of the

monoenergetic case, it is shown that the primary electrons prefer to be forwards ejected. It

has been found that for the various materials and mammographic spectra the percentage of

primary electrons being forwards ejected ranges from 57 % (e.g. TlBr, for x-ray spectrum:

Mo, kVp: 20, HVL: 0.3 mm Al) to 61 % (a-Se, for x-ray spectrum: MoW, kVp: 40,

HVL: 2.03 mm Al, filter: Beam Shaping 0.13 mm of LA) whereas the most probable polar

angle ranges from 0.873 rad (50o) (Ge, for x-ray spectrum: MoW, kVp: 40, HVL: 2.03 mm

Al, filter: Beam Shaping 0.13 mm of LA) to 1.223 rad (70o) (e.g. a-As2Se3, for x-ray

spectrum: Mo, kVp: 20, HVL: 0.3 mm Al).

8.4. Spatial distributions of primary electrons.

For the various materials and spectra studied the two-dimensional (2D) spatial

distributions of primary electrons on the xy and yz planes of the detector have been

87

0

2

4

6

8

10

12

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1cosθ

Num

ber

of p

rim

ary

elec

tron

s x 1

03


calculated. For a better visualization and interpretation of the xy distributions a subregion

of 2 mm width, 2 mm length and 1 mm depth was selected. The pixel size for both xy and

yz distributions is 2 μm 2 μm.

Figure 8.11. (a) The 2D spatial distribution of primary electrons on the detector xy plane

for TlBr at 20 keV. A logarithmic scale in the color depth axis is used. (b) The horizontal

profile histogram (square marks) for the 2D distribution at the point of x-ray incidence

and the corresponding Gaussian fitting curve (solid line).

8.4.1. Monoenergetic case.

The xy distributions of primary electrons are similar to the distribution presented in figure

8.11(a) which is the case of TlBr at 20 keV incident x-ray energy. The point spread

function (PSF) describes the response of a system to a delta-function. In order to study the

energy dependence of the PSF of primary electrons in the various materials, for each 2D

xy distribution a horizontal and a vertical profile histogram have been made at the point of

x-ray incidence. The profiles have a Gaussian shape and are similar to the horizontal

profile presented in figure 8.11(b) (square marks) which is for TlBr at 20 keV incident

x-ray energy. A Gaussian fit was made for each profile histogram that was of the form:

(8.1)

88

0 75 150 -75 -150

75

150

0

-75

-150

x (µm)

y (µm)

(a)

(b)


with α1, b1 and c1 being the fitting parameters. In figure 8.11(b) the Gaussian fit is shown

with the solid line while with 95% probability the fitting parameters values were

α1= (1.836 0.001) 107, b1= (1.108 53.660) 10-4 μm and c1= (0.8756 0.0024) μm.

Provided that for a Gaussian distribution c1= σ, with σ being the standard deviation, the

full width at half maximum (FWHM) at each energy was calculated as 2.35σ.

Furthermore, the energy dependence of the PSF was also studied in terms of the horizontal

and vertical logarithmic profile histograms at the point of x-ray incidence that provide

additional information concerning the primary electron production that takes place away

from the centre of the xy distributions.

The photoelectric absorption of incident photons, followed by the atomic

deexcitation that produces Auger and CK electrons, occurs almost exclusively at the point

of x-ray incidence. In addition, incident photons that are Compton scattered also create

primary electrons at the spot of x-ray incidence. Consequently, the majority

(approximately 80%) of primary electrons are produced at the point of x-ray incidence.

This is seen in figure 8.11 in which the majority of electrons is produced at the central

pixel. Therefore the FWHM is affected only from the number of electrons created in the

first neighbours of the central pixel. When the number of electrons in the first neighbours

increases, the FWHM increases as well. Similarly, when the number of electrons

decreases, so does the FWHM.

As characteristic examples, the FWHM of the fitted PSFs as a function of incident

x-ray energy E for GaSe, CdTe, PbI2 and CdZnTe is given in figures 8.12(a), 8.12(b),

8.12(c) and 8.12(d), respectively. The estimated FWHM values correspond to the current

resolution limit of 2 μm whereas for all materials and incident x-ray energies the

estimation error is of the order of 10-3 μm.

The FWHM in materials like GaSe (a-Se, a-As2Se3, GaAs and Ge) at energies

smaller than the K edges (for example at E 10 keV in figure 8.12(a)) initially increases

and then gradually decreases. This is due to the fact that initially the majority of scattered

photons is absorbed close to the point of x-ray incidence and thus as the probability for

scattering increases the number of electrons increases there as well. At higher energies

though, the scattered photons can be absorbed at greater distances and consequently the

number of electrons close to the point of incidence decreases. In the rest of materials, at

energies smaller than the K edges or the L edges of Hg, Tl and Pb, the FWHM initially

increases but then tends to become constant (figures 8.12(b)-(d)). In PbO this is due to

the limited free path lengths of the scattered photons which, in this way, create electrons

89


close to the point of incidence, while in the rest of materials this is due to the presence of

low energy L fluorescent photons that significantly increase the number of electrons close

to the point of x-ray incidence and decrease the influence of scatter. At the rest of x-ray

energies, the only factor affecting the FWHM in the various materials is the emission of

fluorescent photons. For example in GaSe (figure 8.12(a)) the FWHM increases at

Figure 8.12. The FWHM of the fitted PSFs as a function of incident x-ray energy for

(a) GaSe, (b) CdTe, (c) PbI2 and (d) CdZnTe. The FWHM values correspond to the

current resolution limit of 2 μm whereas the estimation error is of the order of 10-3 μm.

Ga K edge (10.367 keV) and further increases at Se K edge (12.658 keV) because the

absorption of Ga and Se K fluorescent photons increases the number of electrons close to

the point of incidence. In CdTe and PbI2 (figures 8.12(b) and 8.12(c)) the FWHM does not

significantly increase at the K edges of Cd (26.711 keV) and I (33.169 keV) because Cd

and I K fluorescent photons have higher energies and can produce electrons at greater

distances from the point of x-ray incidence. Due to this fact, the FWHM in CdZnTe

(figure 8.12(d)) decreases at Cd K edge. Nevertheless, Te K fluorescent photons are

90

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

FWH

M (

μm)

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

FWH

M (μ

m)

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

FWH

M (μ

m)

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

FWH

M (μ

m)

(a) (b)

(c) (d)


absorbed close to the point of x-ray incidence due to the presence of Cd K edge and thus

the FWHM in figures 8.12(b) and 8.12(d) increases at Te K edge (31.814 keV).

The practical mammographic energy range can be divided into three zones with

respect to the values of FWHM in the various materials: (a) Zone A (Pb BLI = 15.861

keV<E 26.711 keV= Cd BK), (b) Zone B (Cd BK = 26.711 keV<E 31.814 keV= Te BK)

and (c) Zone C (Te BK = 31.814 keV<E 40 keV). Since the FWHM for a material does

Table 8.1: The average values of the FWHM for the various materials in ascending order

in Zone A (Pb BLI = 15.861 keV<E 26.711 keV= Cd BK), Zone B (Cd BK = 26.711 keV<E

31.814 keV= Te BK) and Zone C (Te BK = 31.814 keV<E 40 keV). The FWHM values

correspond to the current resolution limit of 2 μm whereas the estimation error is of the

order of 10-3 μm.

Zone A Zone B Zone CMaterial FWHM

(μm)Material FWHM

(μm)Material FWHM

(μm)a-Se 1.346 a-Se 1.345 a-Se 1.345a-As2Se3 1.372 a-As2Se3 1.372 a-As2Se3 1.372Ge 1.404 Ge 1.404 Ge 1.403CdTe 1.411 CdTe 1.424 TlBr 1.453TlBr 1.455 Cd0.8Zn0.2Te 1.437 PbO 1.474Cd0.8Zn0.2Te 1.470 CdZnTe 1.444 PbI2 1.475PbI2 1.472 TlBr 1.455 HgI2 1.477PbO 1.476 PbI2 1.471 GaSe 1.486HgI2 1.477 HgI2 1.476 CdTe 1.497GaSe 1.485 PbO 1.476 CdZnTe 1.506CdZnTe 1.492 GaSe 1.486 ZnTe 1.507GaAs 1.507 GaAs 1.509 GaAs 1.509ZnTe 1.568 ZnTe 1.567 Cd0.8Zn0.2Te 1.509

not significantly change in a particular zone, in table 8.1 the average values of the FWHM

for the various materials at each zone are shown in ascending order. It is seen that the

FWHM values of CdTe, CdZnTe and Cd0.8Zn0.2Te in Zone A are spread out widely due to

the presence of Zn K fluorescent photons that significantly increase the FWHM especially

in CdZnTe. On the other hand, this is not the case in Zones B and C because in Zone B

the presence of Cd K fluorescent photons does not significantly increase the FWHM in

CdTe but significantly decreases the FWHM in CdZnTe and Cd0.8Zn0.2Te whereas in

Zone C the presence of Te K fluorescent photons significantly increases the FWHM in the

three materials. Furthermore, it is concluded that in the practical mammographic energy

91


range and at this primitive stage of primary electron production, a-Se has the best inherent

spatial resolution as compared to the rest of photoconductors.

As characteristic examples, the horizontal logarithmic profile histograms at the point

of x-ray incidence at various energies for CdZnTe and PbI2 are shown in figures 8.13 and

8.14, respectively. The radius of the spatial distributions (the range from the point of x-ray

incidence in which the primary electrons are produced) is affected from scatter and the

emission of fluorescent photons that can travel away from the point of x-ray

Figure 8.13. The horizontal logarithmic profile histograms at the point of x-ray incidence

for CdZnTe at (a) 2 keV, (b) 9 keV, (c) 10 keV, (d) 26 keV, (e) 27 keV and (f) 40 keV.

92

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

1.00E+06

1.00E+07

1.00E+08

-700 -600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600 700

Horizontal Position (μm)

Num

ber

of p

rim

ary

elec

tron

s

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

1.00E+06

1.00E+07

1.00E+08

-700 -600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600 700Horizontal Position (μm)

Num

ber

of p

rim

ary

elec

tron

s

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

1.00E+06

1.00E+07

1.00E+08

-700 -600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600 700


Num

ber

of p

rim

ary

elec

tron

s

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

1.00E+06

1.00E+07

1.00E+08

-700 -600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600 700


Num

ber

of p

rim

ary

elec

tron

s

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

1.00E+06

1.00E+07

1.00E+08

-700 -600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600 700


Num

ber

of p

rim

ary

elec

tron

s

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

1.00E+06

1.00E+07

1.00E+08

-700 -600-500 -400-300 -200-100 0 100 200 300 400 500 600 700


Num

ber

of p

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elec

tron

s

(a) (d)

(b) (e)

(c) (f)


incidence before being absorbed. For example in CdZnTe the radius at E 9 keV

(figures 8.13(a) and 8.13(b)) increases due to scatter while at 10 keV (figure 8.13(c)) the

emission of Zn K fluorescent photons increases the number of electrons within the

existing range and thus the radius does not change. Therefore up to 26 keV (figure

8.13(d)) the radius increases due to scatter. The same effect have the L fluorescent

photons of Pb in PbI2 at 16 keV (figure 8.14(c)) and thus for energies up to 33 keV

Figure 8.14. The horizontal logarithmic profile histograms at the point of x-ray incidence

for PbI2 at (a) 2 keV, (b) 13 keV, (c) 16 keV, (d) 33 keV, (e) 34 keV and (f) 40 keV.

93

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

1.00E+06

1.00E+07

1.00E+08

-600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600Horizontal Position (μm)

Num

ber

of p

rim

ary

elec

tron

s

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

1.00E+06

1.00E+07

1.00E+08


Num

ber

of p

rim

ary

elec

tron

s

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

1.00E+06

1.00E+07

1.00E+08


Num

ber

of p

rim

ary

elec

tron

s

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

1.00E+06

1.00E+07

1.00E+08


Num

ber

of p

rim

ary

elec

tron

s

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

1.00E+06

1.00E+07

1.00E+08


Num

ber

of p

rim

ary

elec

tron

s

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

1.00E+06

1.00E+07

1.00E+08


Num

ber

of p

rim

ary

elec

tron

s

(a) (d)

(b) (e)

(c) (f)


(figure 8.14(d)) the radius increases due to scatter. At 27 keV in CdZnTe (figure 8.13(e))

and 34 keV in PbI2 (figure 8.14(e)) the emission of Cd and I K fluorescent photons in the

two materials respectively, increases the radius of the profiles because these photons can

be absorbed at large distances. (figure 8.14(d)) the radius increases due to scatter. At 27

keV in CdZnTe (figure 8.13(e)) and 34 keV in PbI2 (figure 8.14(e)) the emission of Cd

Figure 8.15. The yz (depth) distributions of primary electrons for CdZnTe at (a) 2 keV,

(b) 9 keV, (c) 10 keV, (d) 26 keV, (e) 27 keV and (f) 40 keV. The arrow denotes the

incident x-ray beam.

and I K fluorescent photons in the two materials respectively, increases the radius of the

profiles because these photons can be absorbed at large distances. On the other hand, the

emission of Te K fluorescent photons at 32 keV in CdZnTe does not influence the radius

94

0 0.5 1 -0.5 -1

0.5

0

1

y (mm)

z (mm)

(a)

0 0.5 1 -0.5 -1

0.5

0

1

y (mm)

z (mm)

(b)

0 0.5 1 -0.5 -1

0.5

0

1

y (mm)

z (mm)

(c)

0 0.5 1 -0.5 -1

0.5

0

1

y (mm)

z (mm)

(d)

0 0.5 1 -0.5 -1

0.5

0

1

y (mm)

z (mm)

(e)

0 0.5 1 -0.5 -1

0.5

0

1

y (mm)

z (mm)

(f)


because these photons are strongly absorbed from Cd K edge within the existing ranges

and therefore up to 40 keV (figure 8.13(f)) the radius is almost unchanged.

A characteristic example of the yz (depth) distributions of primary electrons is

shown in figure 8.15 that presents the case of CdZnTe at energies (a) 2 keV, (b) 9 keV,

(c) 10 keV, (d) 26 keV, (e) 27 keV and (f) 40 keV. The distributions are explained similar

to the logarithmic profiles of the xy distributions. At energies lower than Cd K edge

(figures 8.15(a)-(d)), as the energy increases the electrons are created deeper inside the

bulk and the distributions become wider as a result of scatter increase. At energies higher

Figure 8.16. The projection of the yz (depth) distribution for CdZnTe at 40 keV. D50%

denotes the depth at which the number of electrons has fallen to half of the value at the

detector’s surface and Dmax the maximum depth at which primary electrons are produced.

than Cd K edge (figures 8.15(e) and 8.15(f)), the emission of fluorescent photons

dominates over the influence of incident energy and scatter and therefore the distributions

remain almost unchanged. For each yz distribution, the corresponding projection has been

calculated. A characteristic example is shown in figure 8.16 that presents the projection of

the yz distribution for the case of CdZnTe at 40 keV.

From the projections the depth at which the number of electrons has fallen to half of

the value at the detector’s surface (D50%) and the maximum depth at which primary

electrons are produced (Dmax) have been calculated. From these projections it has been

calculated that for all the investigated materials and incident energies, the majority of

primary electrons is produced within the first 300 μm from detector’s surface. In table 8.2

95

0

10

20

30

40

50

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Depth (mm)

Gre

y C

olor

D50%

Dmax


the radius of xy distributions (R) as well as the D50% and Dmax are presented for the various

materials at 40 keV. It is important noticing that the Dmax values are actually the minimum

photoconductor thicknesses required. It is seen that at this energy CdTe, CdZnTe and

Cd0.8Zn0.2Te have similar R values because the emitted Cd and Te K fluorescent photons

in these materials are absorbed within the same distance from the point of x-ray

incidence. Furthermore, it is

concluded that PbO has the

minimum bulk space in which

electrons can be produced whereas

CdTe has the maximum one.

Table 8.2: The radius of xy

distributions (R), the maximum

depth at which primary electrons

are produced (Dmax) and the

depth at which the number of

electrons has fallen to half of the value at the detector’s surface (D50% ) for the various

materials at 40 keV. The materials are in ascending order with respect to the values of R.

8.4.2. Polyenergetic case.

The energies of the polyenergetic spectra are higher than the K edges of a-Se, a-As2Se3,

GaSe, GaAs and Ge as well as the L edges of PbO and TlBr. Consequently, in these

materials both the xy and yz spatial distributions are almost the same for all the incident

spectra while the values of FWHM, R, Dmax and D50% are almost constant and similar to

Material R (μm) Dmax (μm) D50% (μm)PbO 200 320 70TlBr 300 400 83Ge 300 500 107GaSe 300 560 105GaAs 310 560 98a-Se 350 540 130a-As2Se3 350 520 126HgI2 370 500 157ZnTe 400 490 157PbI2 400 490 157CdZnTe 450 570 190Cd0.8Zn0.2Te 500 630 192CdTe 500 660 193

96


the values given in tables 8.1 and 8.2. On the other hand, the K edges of Cd, Te and I in

CdTe, CdZnTe, Cd0.8Zn0.2Te, ZnTe, PbI2 and HgI2 exist at higher energies and

therefore the spatial distributions depend on the incident spectrum. For example, the xy

distributions of CdTe for polyenergetic spectra in which the majority of photons have

energies higher than Cd K edge (i.e. the spectrum resulting from W, kVp: 40, HVL: 1.22

mm Al, filter Al: 1.02 mm) have

larger radius compared to the

distributions for spectra in which

the majority of photons have

energies smaller than Cd K edge

(i.e. the spectrum resulting from

Mo, kVp: 30, HVL: 0.5 mm Al, filter Mo: 0.03 mm). Table 8.3 gives the values of R, Dmax

and D50% for these materials for an x-ray spectrum resulting from Mo, kVp: 20, HVL: 0.30

mm Al with no filter, which has the minimum mean energy among all the considered

polyenergetic spectra. Therefore, the values of FWHM are confined within the values

given in table 8.1 whereas R, Dmax and D50% range between the values given in table

8.3 (minimum values) and table 8.2 (maximum values). CdTe,

Table 8.3: The radius of xy distributions (R), the maximum depth at which primary

electrons are produced (Dmax) and the depth at which the number of electrons has fallen to

half of the value at the detector’s surface (D50% ) for CdTe, CdZnTe, Cd0.8Zn0.2Te, ZnTe,

PbI2 and HgI2 for an x-ray spectrum resulting from Mo, kVp: 20, HVL: 0.30 mm Al with no

filter. The materials are in ascending order with respect to the values of R.

CdZnTe and Cd0.8Zn0.2Te have similar R values in table 3 because the scattered photons

in CdTe and the emitted Zn K fluorescent photons in CdZnTe and Cd0.8Zn0.2Te are

absorbed within the same distance from the point of x-ray incidence.

8.5. Arithmetics of photons and primary electrons.

Material R (μm) Dmax (μm) D50% (μm)ZnTe 75 110 33PbI2 100 120 51HgI2 100 120 46CdTe 150 200 51CdZnTe 150 180 50Cd0.8Zn0.2Te 150 200 51

97


For all materials and incident spectra the majority of primary electrons is produced within

the first 300 μm from detector’s surface. Since the typical thickness of the

photoconductors is 500 μm, the results concerning the arithmetics of fluorescent photons,

escaping photons and primary electrons, which have been obtained for 1 mm

thickness, adequately describe the primary signal formation stage. According to their

atomic compositions the materials are grouped into four categories as shown in table 8.4.

Table 8.4. The four categories in which the materials are grouped according to their

atomic compositions.

Figure 8.17. The energy-related number distributions of primary photons that escape

forwards and backwards in (a)

GaSe and (b) CdZnTe. The

number of incident photons

is 107.

8.5.1. Arithmetics of escaping primary photons.

Category MaterialsA a-Se, a-As2Se3, GaSe, GaAs, GeB CdTe, CdZnTe, Cd0.8Zn0.2Te, ZnTeC PbO, TlBrD PbI2, HgI2

98

050

100150200250300350400450500550600

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

Num

ber

of p

rim

ary

phot

ons e

scap

ing

x 10

3

TotalForwardsBackwards

.

05

1015202530354045505560

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

Num

ber

of p

rim

ary

phot

ons e

scap

ing

x 10

3


(a) (b)


In figure 8.17 the energy-related number distributions of primary photons that escape

forwards and backwards in (a) GaSe and (b) CdZnTe are shown as representative results.

The dips, for example at 11 keV in GaSe and 27 keV in CdZnTe, are due to the absorption

edges. In all materials and energies, except for energies E 30 keV in materials of category

A, primary photons escape backwards and their number increases with energy due to the

increase in the probability of scattering. Nevertheless, the escaping percentage is less than

1%. For E 30 keV in materials of category A, for example in GaSe (figure 8.17(a)),

the number of forwards escaping photons increases and exceeds that of those escaping

backwards. The maximum percentage of primary photons that escape is 6% (GaSe at 40

keV) while the average is 0.405%.

8.5.2. Arithmetics of fluorescent photons produced.

In figure 8.18 the energy-related number distributions of fluorescent photons produced in

(a) a-As2Se3 and (b) PbI2 are shown as representative results. The distributions make

jumps at the absorption edges due to the atomic deexcitation. At E 30 keV in materials of

category A, for example in a-As2Se3 (figure 8.18(a)), the number of fluorescent

photons produced slightly decreases due to the increase in the number of primary

photons that escape forwards. In materials of categories B and D, for example in PbI 2

(figure 8.18(b)), there is a slight but gradual increase in the number of fluorescent photons

at energies higher than Cd, Te and I K edges, because the probability of a photon to be

Figure 8.18. The energy-related number distributions of fluorescent photons produced

in (a) a-As2Se3 and (b) PbI2.

99

00.5

11.5

22.5

33.5

44.5

55.5

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

Num

ber

of fl

uore

scen

t pho

tons

pro

duce

d x

106

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

Num

ber

of fl

uore

scen

t pho

tons

pro

duce

d x

106

(a) (b)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

Num

ber

of fl

uore

scen

t pho

tons

pro

duce

d x

106

a-Sea-As2Se3GaSeGaAsGe

0

1

2

3

4

5

6

7

8

9

10

11

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

Num

ber

of fl

uore

scen

t pho

tons

pro

duce

d x

106

CdTeCdZnTeCd08Zn02TeZnTe

0

0.5

1

1.5

2

2.5

3

3.5

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

Num

ber

of fl

uore

scen

t pho

tons

pro

duce

d x

106

PbOTlBr

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

Num

ber

of fl

uore

scen

t pho

tons

pro

duce

d x

106

PbI2HgI2

(a) (b)

(c) (d)


Figure 8.19. The summary graphs of the energy-related distributions of fluorescent

photons produced in materials of (a) category A, (b) category B, (c) category C and

(d) category D.

absorbed from these shells increases, whereas this absorption is followed by long atomic

deexcitation cascades that yield a large number of fluorescent photons. The summary

Figure 8.20. The energy-related number distributions of fluorescent photons that escape

forwards and backwards in (a) TlBr and (b) CdZnTe.

graphs of the energy-related number distributions of fluorescent photons produced in the

materials of the four categories are presented in figure 8.19.

100

050

100150200250300350400450500550600650

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

Num

ber

of fl

uore

scen

t pho

tons

esc

apin

g x

103


0100200300400500600700800900

100011001200130014001500

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

Num

ber

of fl

uore

scen

t pho

tons

esc

apin

g

x 10

3


(a) (b)


8.5.3. Arithmetics of escaping fluorescent photons.

Figure 8.20 presents the energy-related number distributions of fluorescent photons that

escape forwards and backwards in (a) TlBr and (b) CdZnTe, as representative results. In

all materials, fluorescent photons escape backwards. The backwards escaping is due to

three reasons: (i) the fluorescent photon production site is close to the photoconductor’s

surface, (ii) the fluorescent photon emission is isotropical and (iii) fluorescent photons

have relatively low energies. The distributions make jumps at the absorption edges

whereas as the energy increases the number of escaping fluorescent photons decreases

because the primary photon absorption depth increases. The maximum percentage of

fluorescent photons that escape is 30.701% (a-Se at 13 keV) while the average is 7.482%.

8.5.4. Arithmetics of escaping primary and fluorescent photons.

In figure 8.21 the energy-related number distributions of escaping primary and fluorescent

photons in (a) GaSe and (b) PbI2 are shown as representative results. In all materials and

incident energies, except for E 30 keV in materials of category A, the majority of

escaping photons is fluorescent photons. For E 30 keV in materials of category A, the

number of escaping primary photons increases and exceeds that of escaping fluorescent

photons. The summary graphs of the energy-related number distributions of escaping

photons in the materials of the four categories are presented in figure 8.22.

Figure 8.21. The energy-related number distributions of escaping primary and fluorescent

photons in (a) GaSe and (b) PbI2.

101

0100200300400500600700800900

100011001200

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

Num

ber

of e

scap

ing

phot

ons x

10

3

TotalPrimaryFluorescent

0100200300400500600700800900

10001100

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

Num

ber

of e

scap

ing

phot

ons x

10

3

TotalPrimaryFluorescent

(a) (b)

0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

Num

ber

of e

scap

ing

phot

ons

x 10

3


0

300

600

900

1200

1500

1800

2100

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Energy (keV)

Num

ber

of e

scap

ing

phot

ons

x 10

3


0

100

200

300

400

500

600

700

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Energy (keV)

Num

ber

of e

scap

ing

phot

ons

x 10

3

PbOTlBr

0

200

400

600

800

1000

1200

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Energy (keV)

Num

ber

of e

scap

ing

phot

ons

x 10

3

PbI2HgI2

(a) (b)

(c) (d)


Figure 8.22. The summary graphs of the energy-related distributions of escaping photons

in materials of (a) category A, (b) category B, (c) category C and (d) category D.

Figure 8.23. The energy-related number distributions of primary electrons produced in

(a) a-Se and (b) CdZnTe.

8.5.5. Arithmetics of primary electrons produced.

102

02468

10121416182022

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

Num

ber

of p

rim

ary

elec

tron

s pro

duce

d x

106

0

5

10

15

20

25

30

35

40

45

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

Num

ber

of p

rim

ary

elec

tron

s pro

duce

d x

106

(a) (b)


In figure 8.23 the energy-related number distributions of primary electrons produced in

(a) a-Se and (b) CdZnTe are shown as representative results. The distributions make

jumps at the absorption edges due to the primary photon absorption and the atomic

deexcitation. In materials of category A, for example in a-Se (figure 8.23(a)), there is a

gradual increase in the number of electrons at energies higher than the K edges and up to

30 keV. This is due to the decrease in the number of escaping fluorescent photons. At

higher energies the number of electrons decreases as a result of the forward escaping of

primary photons. In the rest of materials, at energies higher than Cd and Te K edges

as well as Pb, Hg and Tl L edges, for example at E 27 keV in CdZnTe (figure 8.23(b)),

the number of electrons increases with energy. This is due to the fact that: (i) the escaping

of fluorescent photons decreases and (ii) the absorption of fluorescent photons is

followed by long atomic deexcitation cascades that yield a large number of electrons. At

lower energies though, for example in the energy range 10-26 keV in CdZnTe, despite the

fact that there is also a decrease in the escaping of fluorescent photons, yet their

absorption is followed by short atomic deexcitation cascades and therefore the number of

electrons is not seriously affected. The summary graphs of the energy-related number

distributions of primary electrons produced in the materials of the four categories are

presented in figure 8.24.

103

0

2

4

6

8

10

12

14

16

18

20

22

24

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

Num

ber

of p

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elec

tron

s pr

oduc

ed x

10

6


0

5

10

15

20

25

30

35

40

45

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

Num

ber

of p

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elec

tron

s pr

oduc

ed x

10

6


0

2

4

6

8

10

12

14

16

18

20

22

24

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

Num

ber

of p

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elec

tron

s pr

oduc

ed x

10

6

PbOTlBr

0

3

6

9

12

15

18

21

24

27

30

33

36

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40Energy (keV)

Num

ber

of p

rim

ary

elec

tron

s pr

oduc

ed x

10

6

PbI2HgI2

(a) (b)

(c) (d)


Figure 8.24. The summary graphs of the energy-related distributions of primary electrons

produced in materials of (a) category A, (b) category B, (c) category C and

(d) category D.

8.5.6. Summary tables.

Table 8.5 presents the average and the maximum percentages of escaping primary and

fluorescent photons. Table 8.6 presents the materials with the minimum and maximum

number of fluorescent photons, escaping photons and primary electrons for the practical

mammographic energy range (16 keV E 40 keV). It has been found that a-Se has the

minimum primary electron production for the mammographic energies.

Table 8.5. The average and maximum percentages of escaping primary and fluorescent

photons. The materials that correspond to the maximum percentages are also shown.

104

Escaping photons Direction

Average Percentage

(%)

Maximum Percentage

(%)Material

PrimaryForwards 0.2 5.5 GaSe 40 keVBackwards 0.2 0.6 CdTe 26 keV

Total 0.4 5.9 GaSe 40 keV

FluorescentForwards 0.02 0.4 a-Se 40 keVBackwards 7.5 30.7 a-Se 13 keV

Total 7.5 30.7 a-Se 13 keV

Primary & Fluorescent

Forwards 0.2 5.7 GaSe 40 keVBackwards 7.7 30.8 a-Se 13 keV

Total 7.9 30.8 a-Se 13 keV


Table 8.6. The materials with the minimum and maximum number of fluorescent photons,

escaping photons and primary electrons in the practical mammographic energy range

(16-40 keV).

CHAPTER 9

A PRELIMINARY STUDY ON FINAL

SIGNAL FORMATION IN a-Se

105

Energy (keV)

Number of fluorescent photons

Number of escaping photons

Number of primary electrons

min max min max min max

16-26 CdTe GaSe, GaAs

CdTe, CdZnTe,

Cd0.8Zn0.2Te, ZnTe

a-Se, a-As2Se3

PbO, a-Se TlBr, GaAs, GaSe, ZnTe

27-31 PbI2, HgI2, ZnTe

CdTe, CdZnTe,

Cd0.8Zn0.2Te

PbI2, HgI2, ZnTe CdTe PbO, a-Se CdTe

32-33 PbI2, HgI2 CdTe PbI2, HgI2 CdTe PbO, a-Se CdTe

34-40 PbO, TlBr CdTe PbO, TlBr CdTe a-Se CdTe


9.1. Introduction

This chapter presents a first approach made to the simulation of the final signal formation

inside a-Se detectors. In this approach, the primary electrons produced inside a-Se are set

in motion in vacuum under the influence of a uniform electric field. Characteristic results

concerning the energy, angular, spatial and time distributions of primary electrons

reaching the detector’s top electrode are presented and discussed. Hence, a primitive study

of the influence of the characteristics of the primary signal on the characteristics of the

final signal is made.

9.2. Mathematical formulation

In this first approach of the simulation of the final signal formation inside a-Se detectors,

two are the basic assumptions made:

i. The primary electrons drift in the vacuum.

106


ii. A uniform electric field of the form is applied. It is set Vtop el=+10 kV

and da-Se=1 mm and hence the applied electric field has a value of 10 V/μm which is

typical for a-Se direct detectors.

The problem that must be solved is schematically illustrated in figure 9.1.

Figure 9.1. Schematical illustration of the primary electron drifting in the vacuum

under the influence of a uniform electric field of the form E=V/l. Ui and Uf are the initial

and final electron velocities in respect and F is the electric force imposed on electrons.

The z-Global System is also shown.

The information concerning the initial energies Ek;i, positions (xi, yi, zi) and

directions (θi, φi) of primary electrons is already known from the simulation of primary

electron production. The quantities being calculated are the final energies Ek;f, positions

(xf, yf, 0) and directions (θf, φf) of electrons reaching the detector’s top electrode.

The Newton’s law is:

(9.1)

whereas the theorem for kinetic energy change is:

(9.2)

where Wel.f is the work of the electric field. The system of equations (9.1) and (9.2) has

been solved yielding the following results for zf=0 (Top electrode):

Energy of primary electrons:

107

VACUUM

z-Global System

F

E

Uf

Uiz

Uiy

Uix

Ui

Top ElectrodeF

z

y

x


(9.3)

Position of primary electrons:

(9.4a)

(9.4b)

(9.4c)

Drifting time of primary electrons:

(9.5a)

(9.5b)

Direction of primary electrons:

(9.6)

(9.7)

9.3. Results and Discussion

As characteristic examples the results of energy, angular, spatial and time distributions of

primary electrons on detector’s top electrode are presented for an x-ray spectrum resulting

from Mo, kVp: 30, HVL: 0.5 mm Al, filter Mo: 0.03 mm.

9.3.1. Energy distribution of primary electrons on top electrode

Figure 9.1 presents the energy distribution of primary electrons on top electrode for an

x-ray spectrum resulting from Mo, kVp: 30, HVL: 0.5 mm Al, filter Mo: 0.03 mm.

108

0

1

2

3

4

5

6

7

8

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Energy(keV)

Num

ber

of p

rim

ary

elec

tron

s x 1

06

Initial energiesEnergies on top electrode


Figure 9.1. The initial energy distribution of primary electrons and the final distribution

on detector’s top electrode for an x-ray spectrum resulting from Mo, kVp: 30, HVL: 0.5

mm Al, filter Mo: 0.03 mm.

It is seen that the electron energy distribution is shifted at slightly higher energies with a

small change in its shape. This was expected since the majority of primary electrons has

been produced close to the detector’s top electrode (at depths <300 μm).

9.3.2. Time distribution of primary electrons on top electrode

Figure 9.2 presents the time distribution of primary electrons on top electrode for an x-ray

spectrum resulting from Mo, kVp: 30, HVL: 0.5 mm Al, filter Mo: 0.03 mm. The majority

of primary electrons is collected from the top electrode at t<5 x 10 -12 s. This was expected

since most of primary electrons are generated within 300 μm depth. The signal (electrical

pulse) has a duration less than 7.2 x 10-11 s.

Figure 9.2. The time distribution (drifting time) of primary electrons on detector’s top

electrode for an x-ray spectrum resulting from Mo, kVp: 30, HVL: 0.5 mm Al,

109

0

20

40

60

80

100

120

140

160

180

200

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

t x 10-11 s

Num

ber

of p

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elec

tron

s

x

103


filter Mo: 0.03 mm.

9.3.3. Spatial distribution of primary electrons on top electrode

In order to produce a comprehensive image of the spatial distribution of primary electrons

on top electrode, a subregion with dimensions 6 mm x 6 mm has been selected. Figure 9.3

presents the xy spatial distribution of primary electrons on top electrode for an x-ray

spectrum resulting from Mo, kVp: 30, HVL: 0.5 mm Al, filter Mo: 0.03 mm. The pixel

size is 3 μm x 3 μm. It is seen that the xy spatial distribution has two opposing lobes

around y=0 as well as a “ring” at approximately 1.945 mm radial distance. As it has been

discussed in chapter 8, the primary electrons prefer to be ejected at two lobes around φ=0

and π. Since the electric field is applied vertically to the xy plane it does not influence the

electron azimuthal distribution a fact that results in the lobes seen in figure 9.3. The “ring”

is due to the Auger electrons which are being ejected isotropically with maximum

ejection probability at θ=π/2. Figure 9.4 presents the horizontal and vertical profile

histograms at the point of x-ray incidence (center of xy distribution) as well as the

corresponding logarithmic profile histograms. It is seen that the profiles have the

maximum number of electrons at the point of x-ray incidence due to the fact that at this

point the majority of primary electrons has been produced. The peaks at radial distance

~1.945 mm in figures 9.4(b) and (d) are due to the “ring” previously discussed. The

110

0 30.6 1.2 1.8 2.4-3 -0.6-1.2-1.8-2.4

0

0.6

1.2

1.8

2.4

3

-0.6

-1.2

-1.8

-2.4

-3

x (mm)

y (mm)

FWHM of the PSF of primary electrons on top electrode is approximately 7.5 μm, which

is 5.5 times larger than the initial FWHM (1.345 μm).


Figure 9.3. The xy spatial distribution of primary electrons on detector’s top electrode for

an x-ray spectrum resulting from Mo, kVp: 30, HVL: 0.5 mm Al, filter Mo: 0.03 mm. A

logarithmic scale in the colour depth axis is used.

Figure 9.5. The initial polar distribution of primary electrons and the final distribution on

detector’s top electrode for an x-ray spectrum resulting from Mo, kVp: 30, HVL: 0.5 mm

Al, filter Mo: 0.03 mm.

111

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

1.00E+06

-3 -2.4 -1.8 -1.2 -0.6 0 0.6 1.2 1.8 2.4 3y (mm)

Num

ber

of p

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elec

tron

s

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

1.00E+06

-3 -2.4 -1.8 -1.2 -0.6 0 0.6 1.2 1.8 2.4 3x (mm)

Num

ber

of p

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s

0

20

40

60

80

100

120

140

160

-3 -2.4 -1.8 -1.2 -0.6 0 0.6 1.2 1.8 2.4 3x (mm)

Num

ber

of p

rim

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elec

tron

s x

103

(b)

(a)

(d)

(c)0

20

40

60

80

100

120

140

160

-3 -2.4 -1.8 -1.2 -0.6 0 0.6 1.2 1.8 2.4 3x (mm)

Num

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of p

rim

ary

elec

tron

s x

103

Figure 9.4. (a) The horizontal and (b) the corresponding logarithmic profile histogram at

the point of x-ray incidence. (c) The vertical and (d) the corresponding logarithmic profile

histogram at the point of x-ray incidence.

0

50

100

150

200

250

0 0.5 1 1.5 2 2.5 3

Polar angle θ (rad)

Num

ber

of p

rim

ary

elec

tron

s x

103 Initial distribution

Distribution on top electrode


9.3.4. Angular distributions of primary electrons on top electrode

As it was previously discussed the primary electron azimuthal distribution is not altered

during their drifting and consequently the distribution remains the same on detector’s top

electrode. Figure 9.5 presents the polar distribution on top electrode. All primary electrons

have polar angles θ>π/2 a fact that was expected since the applied electric field flips the

primary electron directions towards the top electrode. The distribution has maximum at

approximately θ~111o. Finally, figure 9.6 presents a grey scale image representing the

frequency of appearance of the different (θ, φ) pairs.

CHAPTER 10

ELECTRIC FIELD CONSIDERATIONS

IN a-Se

112

Figure 9.6. The grey scale image representing the frequency of appearance of the

different (θ, φ) pairs.

Azimuthal angle φ (rad)

Pola

r an

gle

θ (r

ad)


10.1. Introduction

In direct conversion digital flat panel imagers the x-ray induced charge carriers (electrons

and holes) drift towards the collecting electrodes under the influence of an applied electric

field. In a-Se, holes are more mobile than electrons (at typical electric field values the hole

range (μhτh) is 30 x 10-6 cm2/V whereas that for electrons (μeτe) is 5 x 10-6 cm2/V). Due to

this fact, a-Se direct detectors have a positive high voltage electrode so that electrons

move towards the top electrode and holes towards the active matrix array. In this way

faster signal acquisition is achieved.

It is obvious that the calculation of a realistic electric field is crucial in the

simulation of signal formation in a-Se detectors. The problem that must be solved is the

Laplace’s equation with the proper boundary conditions. The solution can be either an

113


analytical or a numerical one. The numerical solving is usually done by using the so-called

relaxation methods which are based on finite differencing. The main relaxation methods

are the Jacobi’s method, the Gauss-Seidel method and the Succesive Overrelaxation

(SOR) method (detailed information on these methods as well as relevant Fortran codes

can be found in Numerical Recipies in Fortran 77 by Press et al).

Pang et al (1998) calculated the electric field inside an a-Se detector analytically.

Their goal was to make the charge collection by pixel electrodes almost complete by

depositing holes in the pixel gaps. The boundary conditions considered are similar to our

case. Therefore, the calculated electric field is suitable for the modeling of primary

electron drifting inside a-Se. This chapter presents the calculation method of Pang et al

(1998) and additional numerical calculations carried out to obtain the electric potential

distribution anywhere inside a-Se over the pixel and the pixel gap.

10.2. Boundary conditions

Figure 10.1 is a schematic of the simplified cross section considered for an a-Se direct

conversion digital detector to calculate the electric field distribution. The top electrode is

an ITO (Indium Tin Oxide) electrode at a positive bias V top el=5000 V. The a-Se

thickness is da-Se=500 μm, the pixel electrode has a voltage Vp~10 V whereas the active

matrix lays onto a grounded insulating layer (SiO2) that has a thickness dSiO2=1-5 μm. As

holes drift towards the active matrix, some of them land in the gaps between the pixels

contributing in a loss of charge. As the number of holes in the gaps increases, the electric

field is locally inverted and hence some of the trapped holes drift back inside a-Se bulk.

Figure 10.1. A schematic of the simplified cross section considered for an a-Se direct

conversion digital detector to calculate the electric field distribution. The top electrode is

an ITO (Indium Tin Oxide) electrode at a positive bias V top el=5000 V. The a-Se

114

a-Se Active Matrix

Insulator SiO2

ITO Top electrode Vtop el=5000 V

da-Se=500 μm

dSiO2=1-5 μm

Vp~10 V

++++ ++++ ++++ ++++ ++++ ++++

σgap(x,y)

x

z


thickness is da-Se=500 μm, the pixel electrode has a voltage Vp~10 V whereas the active

matrix lays onto a grounded insulating layer (SiO2) that has a thickness dins= 1-5 μm.

Holes land at the gaps between the pixels resulting in a surface density of positive charges

σgap(x,y).

After equilibrium is achieved a constant hole concentration can be considered

in the gaps.

Pang et al solved the Laplace’s equation in three dimensions:

(10.1)

in both the a-Se and the insulator layers with the following boundary conditions:

(10.2)

(10.3)

If (x,y) is on the pixel electrodes

(10.4)

If (x,y) is in the gap region

(10.5)

In the above equations δ denotes the infinitesimal small whereas the quantities εa-Se and

εSiO2 are the dielectric constants of the a-Se and the insulator layers in respect (εa-Se=6.3,

εSiO2=3.8997). In the case that σgap(x,y) is unknown but the field distribution in the gap

region i.e. Ez(x,y) is known, the boundary condition (10.5) should be replaced by:

(10.6)

The assumption is that there is no space charge in a-Se and insulator layers (except at their

interface). The electric field is calculated from .

10.3. Calculation of the electric potential distribution

Figure 10.2(a) presents the front view of the pixel plane at z=da-Se. It is seen that the

geometry is symmetrical and periodical with respect to x=0 and y=0. Hence, Pang et al

115


(1998) calculate the electric potential distribution at a quarter of the whole pixel and at

half the gap width as shown in figure 10.2(b).

Figure 10.2. (a) The front view of the pixel plane at z=da-Se . (b) Due to the periodicity and

the symmetry of the geometry, Pang et al (1998) calculate the electric potential at a

quarter of the whole pixel and at half the gap width (square region).

The expressions derived from the solution of Laplace’s equation with the above boundary

conditions are the following (to simplify the expressions we set da-Se=D, dSiO2=d, Vtopel=V1,

Vp=Vo, εa-Se=ε1 and εSiO2=ε2):

a-Se layer:

(10.7)

Insulator layer:

(10.8)

The coefficients A00, Amn, B00 and Bmn are calculated as follows:

From equation (10.3) the relation below is obtained:

116

Tx

Ty

Tx

Ty

(a) (b)


(10.9)

It is defined:

(10.10)

which satisfies the following equation:

(10.11) The expressions for and depend on whether the boundary

condition (10.5) (σgap(x,y) is known) or (10.6)(Ez(x,y) is known) is used. Therefore:

If σ gap(x,y) is known:

(10.13)

(10.14)

If E z(x,y) is known:

(10.15)

117

(10.12)


(10.16)

(10.17)

Multiplying both sides of equation (10.11) by and integrating

over x and y, it is obtained:

(10.18)

where

(10.19)

From equations (10.18)-(10.20) the parameters A00, Amn, B00 and Bmn are calculated. In this

PhD thesis, equation (10.18) was solved numerically. In particular the truncation

approximation was used to replace in (10.18) by , where Nmax is an integer.

Equation (10.18) was transformed into the matrix equation:

EC=I (10.21)

where E is a (Nmax+1)2 x (Nmax+1)2 matrix and C, I are (Nmax+1)2x 1 matrices. The system

(10.21) is solved using the Gauss-Jordan Elimination method (Numerical Recipies in

Fortran 77, Press et al). Once Cmn is known the potential in the a-Se layer can be

calculated from equations (10.7) and (10.10) as (Pang et al 1998):

(10.22)

where

118

(10.20)


(10.23)

for smoothing the Gibbs oscillations caused by the truncation approximation. Figure 10.4.

presents a snapshot during the convergence on the solution for the potential

distribution V(x, y, z) at the pixel plane (z=D) for the case that Ez(x,y)=0 in the gap

region. The input parameters are: Vtop el= 5000 V, da-Se=500 μm, dSiO2= 5 μm, Vp= 10 V,

Tx=Ty= 50 μm, τx=τy=45 μm and Nmax=50.

Figure 10.4. A snapshot during the convergence on the solution for the potential

distribution V(x, y, z) at the pixel plane (z=D) for the case that E z(x,y)=0 in the gap

region. The input parameters are: Vtop el= 5000 V, da-Se=500 μm, dSiO2= 5 μm, Vp= 10 V,

Tx=Ty= 50 μm, τx=τy=45 μm and Nmax=50.

119


CHAPTER 11MODEL FORMULATION FOR

ELECTRON INTERACTIONS IN a-Se

120


11.1. Introduction

The x-ray induced primary electrons inside the photoconductor’s bulk comprise the

primary signal that propagates in the material and forms the final signal (image) at the

detector’s electrodes. As the signal propagates, electrons interact with the material and are

subject to recombination and trapping. Lachaine, Fallone and Fourkal have dealt with the

signal propagation inside a-Se. In particular, Lachaine and Fallone (2000a, b) made

calculations on the electron inelastic scattering cross-sections as well as Monte Carlo

simulations of x-ray induced recombination. Fourkal et al (2001) made a complete

simulation of the signal formation in a-Se. The formulations were based on theoretical

calculations mainly developed by Ashley (1988), La Verne and Pimblott (1995), Pimblott

et al (1996), Green et al (1988), Ritchie (1959) and Hamm et al (1985).

During this PhD thesis the model of Fourkal et al (2001) has been reexamined and

enriched with existing theoretical considerations and simulation formalisms. This chapter

presents the structure and the mathematical formulation of a model that would simulate

the electron interactions inside a-Se.

11.2. Electron free path length

The free path length between two successive electron interactions is assumed to obey

Poisson statistics. Thus the probability density function for the free path length s is:

(11.1)

where λtot is the total mean free path. The number of molecules (atoms) per unit volume is:

(11.2)

where NA is the Avogadro’s number, AM is the molecular weight and ρ is the density. The

electrons are assumed to undergo only elastic and inelastic scattering. Thus, the total

interaction cross section is defined as:

(11.3)

where σel and σinel are the elastic and inelastic scattering cross sections respectively. Thus,

the mean free path is defined as:

121


(11.4)

11.3. Decision on the type of electron interaction

From equation (11.3) it is derived that . If the probabilities for elastic (Pel)

and inelastic (Pinel) scattering are defined as:

(11.5a)

(11.5b)

then a random decision is made based on Pel and Pinel to determine the type of electron

interaction process.

11.4. Elastic scattering

11.4.1. Differential cross section

The theory of elastic scattering has been discussed in chapter 5 and section 5.4.1. Since we

work in non-relativistic electron energies, where the exchange and polarization effects are

negligible, the Mott differential cross section (5.12a) can be written as (Salvat et al 1985):

(11.6a)

where

(11.6b)

As it is stated by Salvat et al (1985) the scattering amplitude can be calculated by

using the first Born approximation and some additional concepts to compensate for the

fact that the Born cross section is not valid for small electron energies. Within the range of

validity of the Born approximation, that is for relatively large energies of the incident

electron (500 eV-50 keV), the Born (B) scattering amplitude is given by (for a measuring

system with m=e=1):

122


(11.7)

where is the momentum transfer. Equation (11.7) can be written as:

(11.8a)

with the phase shifts:

(11.8b)

where are spherical Bessel functions. Salvat et al (1985) assume that:

(11.9a)

with

(11.9b)

If an analytical screened Coulomb potential is assumed of the form:

(11.10)

where A, α1 and α2 are constants that characterize the material, then equation (11.8a)

becomes:

(11.11)

and the phase shifts become:

(11.12)

where Ql are the Legendre functions of the second kind. Therefore, from equation (11.6a)

using equations (11.9), (11.11) and (11.12) and additional calculating ideas Salvat et al

(1985) calculate the elastic scattering cross section.

For the sake of simplicity, it can be assumed that even for low electron energies the

Born approximation is valid and therefore (11.9a) becomes . Using this

assumption equation (11.6a) can be written as:

123


(11.13)

Taking into account that and that for the non-relativistic case we

get:

(11.14)

with and A = 0.4836, α1 =8.7824, α2 =1.6967 for a-Se (Salvat et al 1987). The

elastic scattering angle θ of the electron can be sampled from equation (11.14) using the

rejection method.

11.4.2. Elastic scattering cross section (σel)

The elastic scattering cross section is given by:

(11.15)

Using equation (11.11), and dΩ=2πsinθdθ we calculate that:

(11.16)

Since when θ=0, q2=0 and when θ=π, q2=4k2. Thus equation (11.15) is

written as:

(11.17)

124


Calculating the integral we find that:

with .

11.5. Inelastic scattering

11.5.1. Inelastic scattering with inner shells (K and L shells)

Fourkal et al (2001) state that the inelastic scattering events with inner shells are not

affected by the physical state of the medium. Therefore, they use tabulated cross sections

for independent Se atoms from the Evaluated Electron Data Library (EEDL) of the

Lawrence Livermore National Laboratory (Cullen 2000, Perkins et al 1991). The EEDL:

i. Gives the subshells ionization cross sections.

ii. Gives the energy of the ejected secondary electrons.

iii. Assumes that the direction of the incident electron is not changed during the

interaction process. Thus angular distributions are not given.

iv. Angular distributions of the secondary electrons are not given.

Salvat et al (2003) state that during an inelastic scattering event with inner shell the

correlation between energy loss/scattering of the projectile and ionization events is of

minor importance and may be neglected. Consequently, the inner-shell ionization is

considered as an independent interaction process that has no effect on the state of the

projectile. Accordingly, in the simulation of inelastic collisions with inner shells the

projectile is assumed not to be deflected from its original direction but only cause the

ejection of knock-on electrons (delta rays).

From what it is mentioned above, it is obvious that the only quantity that must be

calculated is the energy loss W of the incident electron. Salvat et al (1987) have calculated

the differential cross section for inelastic collisions with inner shells using a

semiphenomenological approach. In this approach the relationship between the optical

oscillator strength (OOS) of ith inner shell with the photoelectric cross section for

absorption of a photon with energy W from this shell, σph,i(Z,W), is:

125

(11.18)


(11.19)

This relationship holds when the dipole approximation is applicable i.e. when the

wavelength of the photon is much larger than the size of the active shell. Following the

formalism of Salvat et al (2003), the generalized oscillator for ith inner shell is:

(11.20a)

with

(11.20b)

(11.20c)

and Θ being the step function, Zi is the number of electrons of ith shell and Bi is the

binding energy of the inner shell. Using equation (11.20a) Salvat et al calculated the

differential cross section for inelastic scattering with inner shells, which for the case of

non-relativistic energies is:

where (11.21b)

(11.21c)

Consequently the steps that must be followed to calculate the differential cross section are:

i. Calculation of from (11.19).

ii. Setting the number of electrons in the ith subshell, Zi.

iii. Calculation of the integral by making a fit to the data of

photoelectric cross section and integrating analytically.

iv. Rejection method to sample the energy loss W of the incident electron.

The Born approximation overestimates the differential cross sections for incident

electrons with kinetic energies near the binding energy Bi. This is mainly due to the

126

(11.21a)


distortion of the projectile wave function by the electrostatic field of the target atom. To

account for this effect we assume that the incident electron gains a kinetic energy 2Bi and

that Wmax=(E+Bi)/2. The inelastic scattering cross section with inner shells is given by:

(11.22)

The Coulomb correction reduces the differential cross section near the threshold B i and

yields values in better agreement with the experimental data.

11.5.2. Inelastic scattering with outer shells

The model of Fourkal et al (2001) for simulating the inelastic collisions of electrons with

outer shells is based on a theory developed by Ashley (1988). Some comments for this

model are given below:

i. It is a semi-empirical one and describes the inelastic interactions of low energy

electrons with condensed matter in terms of the optical properties of the considered

medium.

ii. It is a statistical model: the stopping medium is viewed as an inhomogeneous

electron gas and the differential inverse mean free path (DIMFP) is obtained as an

average of the DIMFPs in free electron gases of different densities. Weights are used

to average the free electron gas’s DIMFPs with the incorporation of experimental

optical dielectric data.

iii. It is not a relativistic one.

iv. Ashley uses experimental OOSs and accounts for exchange effects.

v. The model leads to realistic results for low energy electrons i.e. when the majority of

excitations correspond to the outer shells. The model is not suitable for describing

inner-shell ionizations.

The complex dielectric function (q,w) gives the response of a medium to a given

energy transfer W and momentum transfer q. The medium is assumed to be homogeneous

and isotropic so that is a scalar quantity and not a tensor. The probability of an energy

loss W per unit distance traveled by a non-relativistic electron of energy E is (in

atomic units i.e. =m=e=1):

(11.23)

127


with . This expression for assumes that the energy-momentum

transfer relation for the electron moving in the medium is the same as that for a free

electron in vacuum. The extension of the energy-loss function to q>0 from the optical

limit is made through:

(11.24)

The energy loss sum rule is:

(11.25)

with no being the density of atoms or molecules in the medium with Z electrons per atom

or molecule. The quantity is called “binding energy”, but it has nothing to do with the

binding energy of electrons in atomic shells. Its meaning will be discussed later on.

Equation (11.23) using (11.24) becomes:

(11.26)

with

(11.27)

Equation (11.26) can be rewritten including exchange effects and indistinguishability as:

(11.28)

The exchange effects concern spin interactions. The indistinguishability can be understood

as follows: an energy transfer W by the primary electron reducing its energy to E-W gives

an electron which cannot be distinguished from the secondary electron of energy E-W

128


produced by a different energy transfer from the primary electron to the struck electron.

Figure 11.1 illustrates this situation schematically.

Figure 11.1. Schematical illustration of the indistinguishability between a scattered

projectile with energy E1-W and a secondary electron (δ-ray) with the same energy.

The Ashley’s approximation (11.24) can be rewritten as follows (in S.I):

where f(B) is the OOS, is the energy loss and is the previously

mentioned binding energy. The physical meaning of B, that is of , can be understood

from the following equation:

(11.30)

where the term is the kinetic energy of a free and initially at rest electron that

acquires momentum q. The region of integration on ( , B) plane is formed from the

following constrains:

(Energy conservation)

(Momentum conservation) (11.31)

129

δ-ray

E1-WE1

δ-rayEδ=E1-W

E2-E2

(11.29)


Therefore the differential cross section for energy loss in inelastic scattering with an

outer shell is derived from equations (11.28), (11.29) and (11.31):

(11.32a)

with (11.32b)

The inelastic scattering cross section for energy loss is:

(11.33a)

Where

(11.33b)

(11.33c)

and F is the incomplete elliptic integral of first kind F(φ,k)= .

130

(11.32c)

CHAPTER 12: GENERAL DISCUSSION, CONCLUSIONS & FUTURE WORK 131

CHAPTER 12: GENERAL DISCUSSION, CONCLUSIONS & FUTURE WORK

CHAPTER 12

GENERAL DISCUSSION,

CONCLUSIONS & FUTURE WORK

132


12.1. General discussion

Despite the fact that film-screen mammography is still the gold standard in the

examination of the female breast, its dynamic range is limited (1:25) whereas masses and

microcalcifications, important indicators of cancer, are hardly visualized in very dense

breasts. Direct conversion digital flat panel mammographic detectors offer the advantages

of digital technology, namely the flexible image acquisition, processing and storage, as

well as wider dynamic range, increased quantum efficiency, reduced blurring and high

spatial resolution. In trying to increase the sensitivity and specificity of the diagnostic

procedure, an important research field deals with the optimization of image quality and the

minimization of dose in breast with the refinement and better design of such systems.

In direct detectors, a photoconductor directly converts the incident x-rays to a charge

cloud that is electrically driven and stored in the pixels. Therefore, the photoconducting

material is one of the most important components. Materials such as a-Se, a-As2Se3, GaSe,

GaAs, Ge, CdTe, CdZnTe, Cd0.8Zn0.2Te, ZnTe, PbO, TlBr, PbI2 and HgI2 satisfy some of

the characteristics of the ideal case for these systems. To improve the image quality and

hence the diagnostic information acquired, a careful selection of the photoconducting

material must be made with the simultaneous optimization of detector technology. These

can be achieved with the investigation of the physics that governs the signal formation

processes in the photoconductors mentioned since in this way important information

relevant to the production of the final image is acquired.

The quality of the mammographic image is directly related to its characteristics.

The x-ray induced primary electrons inside the photoconductor’s bulk comprise the

primary signal which propagates in the material and forms the final signal (image) at the

detector’s electrodes. Consequently, the characteristics of the mammographic image

strongly depend on the characteristics of the primary electrons. The experimental research

is not able to study exclusively the primary electrons. On the other hand, despite the fact

that there is a number of commercially available Monte Carlo simulation packages such as

EGS4 and PENELOPE that deal with photon and electron transport, simulation studies

have not dealt with the characteristics of primary electrons such as their number as well as

their energy, angular and spatial distributions and furthermore with their influence on the

characteristics of the final image.

In this PhD thesis an investigation has been carried out concerning the primary

signal formation processes and the characteristics of primary electrons inside the

photoconducting materials mentioned. In addition, the influence of the characteristics of

133


primary electrons on the characteristics of the final signal together with the electric field

distribution and the electron interaction mechanisms particularly for the case of a-Se, one

of the most preferable photoconductors, have been studied at a first stage. The electric

field distribution and the electron interactions are two crucial parameters in the

development of a model that would simulate the final signal formation and hence study

the influence of the characteristics of the primary electrons on the characteristics of the

final image.

In particular, a Monte Carlo model that simulates the primary electron production

inside a-Se, a-As2Se3, GaSe, GaAs, Ge, CdTe, CdZnTe, Cd0.8Zn0.2Te, ZnTe, PbO, TlBr,

PbI2 and HgI2 has been developed. The model simulates the primary photon interactions

(photoelectric absorption, coherent and incoherent scattering), as well as the atomic

deexcitations (fluorescent photon production, Auger and Coster-Kronig electron

emission). The development of the model was based on a cost versus benefit approach

regarding the accuracy of the results and the algorithmic simplicity i.e. feasible program

execution time.

The obtained results concern the energy and the number of fluorescent photons,

escaping photons and primary electrons, as well as the angular and spatial distributions of

primary electrons. They have been obtained for 107 x-ray photons which are incident

vertically at the center of a detector with dimensions 10 cm width, 10 cm length and 1 mm

thickness, as well as 39 monoenergetic spectra, with energies between 2 and 40 keV, and

53 mammographic spectra, in which the majority of photons has energies between 15 and

40 keV.

In addition, a mathematical formulation has been developed for the drifting of

primary electrons of a-Se in vacuum under the influence of a capacitor’s electric field and

the resulting electron energy, angular and spatial distributions on the collecting electrode

have been studied. The formulation has been based on the Newton’s equations of motion

and the theorem for kinetic energy change.

Furthermore, the electric field distribution of Pang et al (1998) for a-Se detectors has

been adopted and reexamined to adjust it to the simulation model of primary electrons. A

code has been developed that calculates the distribution of the electric potential

anywhere inside a-Se over the pixel and the pixel gap, using the analytical solution of

Pang, the boundary values of our case and the Gauss-Jordan Elimination method.

Finally, the structure and the mathematical formulation of a model that would

simulate the electron interactions inside a-Se have been developed. They were based on

134


the model of Fourkal et al (2001) that has been reexamined and enriched with existing

theoretical considerations, developed mainly by Ashley (1988), and simulation

formalisms, developed mainly by Salvat et al (1985, 1987, 2003).The formulation has

included the electron free path length, the decision on the type of electron interaction, the

differential and total elastic scattering cross section and the differential and total inelastic

scattering cross sections with inner shells (K and L shells) as well as with outer shells.

It has been found that for all materials and energies the energy distributions of

backwards escaping primary photons resemble the shape of the incident spectrum, while

this is not the case for primary photons that escape forwards. The forwards escaping

primary photons have relatively high energies and well above the absorption edges.

Furthermore, the characteristic feature in the primary electron energy distributions for PbI2

and HgI2 is the atomic deexcitation peaks. Since the photoelectric absorption is the

dominant interaction mechanism between x-rays and matter in the mammographic energy

range, the primary electrons are consisted of photoelectrons, Auger and CK electrons.

Therefore, the deexcitation peaks consist of photoelectrons produced by the absorption of

fluorescent photons as well as of Auger and CK electrons. For the rest of materials the

photoelectrons produced from primary photon absorption can also influence the shape of

the distributions. In particular, they give a shape similar to the shape of the incident

spectrum yet shifted at lower energies.

The primary electrons prefer to be ejected forwards. In the mammographic energy

range, the percentage of electrons being forwards ejected is approximately 60 % with the

most probable polar angle ranging from 50o to 70o. In addition, the electrons prefer to be

emitted at two lobes around φ=0 and φ=π. On the other hand, they have the minimum

probability to be ejected at φ=π/2 and 3π/2 and parallel to the incident beam’s axis either

forwards or backwards. The azimuthal uniformity is one of the parameters that define, in

the presence of an electric field, the trajectories of primary electrons in the bulk and

consequently is a factor that affects the final image characteristics. The presence of Auger

and CK electrons increases the azimuthal uniformity, which means smaller tendency of

electron ejection at φ=0, π and 2π. This is due to the fact that these electrons are

isotropically ejected in space. At the practical mammographic energies (15-40 keV) a-Se,

a-As2Se3 and Ge have the minimum azimuthal uniformity whereas CdZnTe, Cd0.8Zn0.2Te

and CdTe the maximum one.

Approximately 80% of primary electrons are produced at the point of x-ray

incidence for all the investigated materials. This is due to the fact that the photoelectric

135


absorption of incident photons, followed by the atomic deexcitation that produces Auger

and CK electrons, occurs almost entirely at the point of x-ray incidence while at the same

time the incident photons that are Compton scattered also create primary electrons at the

spot of x-ray incidence. Both xy (at detector’s plane) and yz (at detector’s depth) electron

spatial distributions are affected from scatter and the emission of fluorescent photons.

The distributions for a-Se, a-As2Se3, GaSe, GaAs, Ge, PbO and TlBr are almost

independent on the polyenergetic spectrum, since their absorption edges have relatively

small energies, while those for CdTe, CdZnTe, Cd0.8Zn0.2Te, ZnTe, PbI2 and HgI2 have a

spectrum dependence, since some absorption edges have higher energies. In the practical

mammographic energy range and at this primitive stage of primary electron production,

a-Se has the best inherent spatial resolution as compared to the rest of photoconductors.

This fact can be evidence that the resolution properties of a-Se are superior. For all the

investigated materials and incident energies, the majority of primary electrons is produced

within the first 300 μm from detector’s surface. PbO has the minimum bulk space in

which electrons can be produced (a radius R=200 μm and a depth Dmax=320 μm) whereas

CdTe has the maximum one (R=500 μm and Dmax=660 μm).

At the stage of primary signal formation and for the typical detector thicknesses

(300-1000 μm), the average fraction of incident x-ray energy transferred to primary

electrons is 97% whereas the minimum is 84.5% (CdTe at 32 keV). The maximum

percentage of fluorescent photons that escape is 30.701% (a-Se at 13 keV) while the

average is 7.482%. The corresponding values for escaping primary photons are 6% (GaSe

at 40 keV) and 0.405%. In all materials and incident energies, except for E 30 keV in a-

Se, a-As2Se3, GaSe, GaAs and Ge (light materials), photons escape backwards whereas the

overwhelming majority is fluorescent photons. The escaping of fluorescent photons and

the atomic deexcitation are the factors that affect the primary electron production. The

number of primary electrons increases at energies higher than the K edges of light

materials, Cd and Te K edges as well as Pb, Hg and Tl L edges where the fluorescent

photon escaping decreases and their absorption is followed by long atomic deexcitation

cascades. For E 30 keV in the light materials, the number of forwards escaping photons

increases, due to the escaping primary photons, and becomes higher than the number of

photons that escape backwards. Furthermore, the primary electron production is

additionally affected by the escaping of primary photons that decreases the number of

electrons. a-Se has the minimum number of primary electrons produced in the practical

mammographic energy range.

136


The results concerning the a-Se primary electrons that have drifted in vacuum under

the influence of a capacitor’s electric field and have reached the collecting electrode

(top electrode) gave a first glimpse at the influence of the characteristics of the primary

signal on the characteristics of the final image. The electron energy distributions are

shifted at slightly higher energies with a small change in their shape. This was expected

since the majority of primary electrons has been produced close to the detector’s top

electrode (at depths <300 μm). The immediate consequence of this fact is that most of

primary electrons are collected at t<5 x 10-12 s whereas the signal (electrical pulse) has a

duration less than 7.2 x 10-11 s. Due to the fact that 80 % of primary electrons is produced

at the point of x-ray incidence, the majority of electrons is collected at this point. The

FWHM of the PSF of primary electrons on top electrode is approximately 5.5 times larger

than its initial value. The xy spatial distributions have two opposing lobes around y=0 as

well as a ring of an approximate radius of 2 mm. The two lobes result from the fact that

the applied electric field is perpendicular to the detector and hence the azimuthal angular

distributions of primary electrons are not affected during their drifting. The ring is due to

the Auger electrons that are isotropically ejected. Finally, all electrons have polar angles

θ>π/2 with the most probable polar angle being θ=1.92 rad ~ 111o.

12.2. Conclusions and future work

The investigation of primary signal formation inside suitable photoconductors for direct

conversion digital flat panel x-ray image detectors has dealt with the number as well as

with the energy, angular and spatial distributions of primary electrons for a number of

monoenergetic and polyenergetic x-ray spectra that cover the mammographic energies.

In this way, insights were gained into the related physics that led to the investigation

of the primary electron characteristics, that strongly influence the characteristics of the

final image, as well as the factors which affect them. The information obtained allows to

make a preliminary choice of the most suitable materials for this kind of applications.

Since TlBr, GaAs, GaSe, ZnTe and CdTe have the maximum number of primary electrons

and high x-ray sensitivity (W ~6 eV), they could be the best choice for high signal gain-

amplification. On the other hand, a-Se, a-As2Se3 and Ge have the best intrinsic spatial

resolution and the minimum azimuthal uniformity. Given that at the presence of an

applied electric field small azimuthal uniformity means degradation of the spatial

resolution mainly at one dimension, these materials could be the best choice for high

spatial resolution. Due to the fact that PbO has the minimum depth at which primary

137


electrons are produced (Dmax=320 μm), it is the best choice for the minimum

photoconductor thickness required. Finally, PbI2 and HgI2 could be the best choices

combining all the required characteristics since:

i. They have high x-ray sensitivity (W ~4.5 eV) and average number of primary

electrons.

ii. The azimuthal uniformity, spatial resolution and the minimum photoconductor

thickness required range on the average.

In contrast with the commercially available simulation packages (EGS4,

PENELOPE etc), that make use of subshell photoelectric cross sections to sample the shell

that absorbs the incident photons and follow the deexcitation mechanisms until the

vacancies have migrated to the outermost shells, the assumptions made in the development

of the model simplify the photoelectric absorption and the atomic deexcitation

mechanisms. Hence, the calculation time is kept under acceptable levels (<20 min. for

a Pentium 4, 2.8 GHz, 448 MB RAM) whereas the validity of the derived results is

preserved.

The results obtained for a-Se primary electrons that have drifted in vacuum under the

influence of a capacitor electric field and have been collected from the top electrode,

although they pertain to an unrealistic case, yet give a first idea of the influence of the

characteristics of the primary signal on the characteristics of the final signal.

Nevertheless, a complete simulation of the signal propagation inside the photoconductor

bulk should be developed in order to derive conclusive remarks on the correlation of

primary and final signal characteristics that would help optimize the performance of direct

detectors and select the most suitable materials for this kind of applications. The basis of

developing such a simulation model can be found on the formulations presented for the

electric field distribution and the electron interactions inside a-Se. Νevertheless, the

formulation for electron interactions inside a-Se needs to be further refined with respect to

its physics in order to develop a simulation model enriched with charge transport and

recombination-trapping mechanisms.

As a future work the formalism presented for the electric field distribution inside

a-Se detectors, will form the basis for the calculation of a realistic electric field inside all

the materials mentioned. The formulation presented to simulate the electron interactions

inside a-Se will be further refined with respect to its physics and a complete simulation

model of the signal propagation inside photoconductor’s bulk will be developed, that will

include charge carrier mobilities and lifetimes, trapping and recombination mechanisms as

138


well as charge transport mechanics. After this stage the model will be able to be validated

with standard experimental techniques. In this way, the dependence of the characteristics

of the final image on the characteristics of the primary electrons will be investigated. This

can contribute in a better selection of the suitable photoconductors for direct conversion

digital flat panel x-ray image detectors as well as in the refinement of their technology.

139

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147

ABSTRACT

ABSTRACT

The quality of the mammographic image is directly related to its characteristics. The

x-ray induced primary electrons inside the photoconductor of direct conversion digital flat

panel mammographic detectors, comprise the primary signal which propagates in the

material and forms the final signal (image). Consequently, the characteristics of the

mammographic image strongly depend on the characteristics of the primary electrons. In

this PhD thesis an investigation is carried out concerning the primary signal formation

processes and the characteristics of primary electrons inside a-Se, a-As2Se3, GaSe, GaAs,

Ge, CdTe, CdZnTe, Cd0.8Zn0.2Te, ZnTe, PbO, TlBr, PbI2 and HgI2, which are suitable

photoconducting materials for direct detectors. In addition, for the case of a-Se a first

study is made concerning the correlation between the characteristics of primary and final

signal as well as the electric field distribution and the electron interaction mechanisms,

two crucial parameters of a prospective model that would simulate the final signal

formation.

A Monte Carlo model that simulates the primary electron production inside the

photoconductors mentioned, for a number of monoenergetic and polyenergetic x-ray

spectra that cover the mammographic energies, has been developed. The model simulates

the primary photon interactions (photoelectric absorption, coherent and incoherent

scattering), as well as the atomic deexcitations (fluorescent photon production, Auger and

Coster-Kronig electron emission). In addition, a mathematical formulation has been

developed for the drifting of primary electrons of a-Se in vacuum under the influence of a

capacitor’s electric field and the electron characteristics on the collecting electrode are

being studied. The formulation is based on the Newton’s equations of motion and the

theorem for kinetic energy change. Furthermore, a code has been developed that calculates

the distribution of the electric potential inside a-Se, using an existing analytical solution,

the boundary values of our case and certain numerical calculation methods. Finally, the

structure and the mathematical formulation of a model that would simulate the electron

interactions inside a-Se have been developed. An existing model has been reexamined and

enriched with certain theoretical considerations and simulation formalisms.

It has been found that for all materials and energies the energy distributions of

backwards escaping primary photons resemble the shape of the incident spectrum, while

this is not the case for primary photons that escape forwards. Furthermore, the

characteristic feature in the primary electron energy distributions for PbI2 and HgI2 is the

148

ABSTRACT

atomic deexcitation peaks. For the rest of materials the photoelectrons produced from

primary photon absorption can also influence the shape of the distributions. The primary

electrons prefer to be ejected forwards. In the mammographic energy range, the

percentage of electrons being forwards ejected is approximately 60 % with the most

probable polar angles ranging from 50o to 70o. In addition, the electrons prefer to be

emitted at two lobes around φ=0 and φ=π. At the practical mammographic energies

(15-40 keV) a-Se, a-As2Se3 and Ge have the minimum azimuthal uniformity whereas

CdZnTe, Cd0.8Zn0.2Te and CdTe the maximum one. The electron spatial distributions are

affected from scatter and the emission of fluorescent photons. The distributions for a-Se,

a-As2Se3, GaSe, GaAs, Ge, PbO and TlBr are almost independent on the polyenergetic

spectrum while those for CdTe, CdZnTe, Cd0.8Zn0.2Te, ZnTe, PbI2 and HgI2 have a

spectrum dependence. In the practical mammographic energy range and at this primitive

stage of primary electron production, a-Se has the best inherent spatial resolution. For all

the investigated materials and incident energies, the majority of primary electrons is

produced within the first 300 μm from detector’s surface. PbO has the minimum bulk

space in which electrons can be produced whereas CdTe has the maximum one. In all

materials and incident energies, except for E 30 keV in a-Se, a-As2Se3, GaSe, GaAs and

Ge (light materials), photons escape backwards whereas the overwhelming majority is

fluorescent photons. The escaping of fluorescent photons and the atomic deexcitation are

the factors that affect the primary electron production. The number of primary electrons

increases at energies higher than the K edges of light materials, Cd and Te K edges as well

as Pb, Hg and Tl L edges where the fluorescent photon escaping decreases and their

absorption is followed by long atomic deexcitation cascades. For E 30 keV in the light

materials, the number of forwards escaping photons increases, due to the escaping primary

photons, and becomes higher than the number of photons that escape backwards.

Furthermore, the primary electron production is additionally affected by the escaping of

primary photons that decreases the number of electrons. a-Se has the minimum number of

primary electrons produced in the practical mammographic energy range. The energy

distributions of primary electrons of a-Se that reach the collecting electrode are shifted at

slightly higher energies with a small change in their shape. Most of electrons are collected

at t<5 x 10-12 s. The majority of electrons is collected at the point of x-ray incidence

whereas the xy spatial distributions have two opposing lobes around y=0 as well as a ring

of an approximate radius of 2 mm. The azimuthal angles are not affected by the electron

149

ABSTRACT

drifting while all electrons have polar angles θ>π/2 with the most probable polar angle

being θ=1.92 rad ~ 111o.

Conclusively, insights are gained into the physics of primary electron production that

lead to the investigation of the primary electron characteristics, which strongly influence

the characteristics of the final image, and the factors which affect them. The results that

concern the electron characteristics on the collecting electrode for the case of a-Se give, at

a first approximation, the dependence of the characteristics of the final signal on the

characteristics of the primary signal. Nevertheless, a complete simulation of the signal

propagation inside the photoconductor’s bulk should be developed in order to derive

conclusive remarks on the correlation of primary and final signal characteristics. The basis

of developing such a simulation model can be found on the formulations presented for the

electric field distribution and the electron interactions inside a-Se.

150

ABSTRACT

ΕΚΤΕΝΗΣ ΠΕΡΙΛΗΨΗ

Η μαστογραφία είναι μέχρι και σήμερα η πιο σημαντική και ευρέως διαδεδομένη

απεικονιστική τεχνική του γυναικείου στήθους. Η μαστογραφική εικόνα πρέπει να είναι

ικανή όχι μόνο να αποκαλύπτει πολύ μικρές διαφορές σύνθεσης και πυκνότητας ιστού,

αλλά ταυτόχρονα την παρουσία των αποτιτανώσεων οι οποίες έχουν ένα τυπικό μέγεθος

γύρω στα 100 μm. Είναι εμφανές ότι είναι απαραίτητο τόσο η αντίθεση εικόνας όσο και η

διακριτική ικανότητα να διατηρούνται σε υψηλά επίπεδα ενώ ταυτόχρονα ο θόρυβος να

παραμένει περιορισμένος. Επιπρόσθετα, λόγω των κινδύνων οι οποίοι υπάρχουν κατά την

χρήση ιοντιζουσών ακτινοβολιών, η δόση στο μαστό πρέπει να είναι τόσο χαμηλή όσο

λογικά μπορεί να επιτευχθεί (ALARA).

Στην προσπάθεια να πραγματοποιηθούν οι παραπάνω αντικειμενικοί στόχοι όπως

επίσης να βελτιωθεί η ευαισθησία και ειδικότητα της μαστογραφικής διαδικασίας,

γεγονός το οποίο θα επέτρεπε μία ποιό ακριβής και ποιό έγκαιρη διάγνωση του καρκίνου

του μαστού, η έρευνα εστιάζει: (α) στη λεγόμενη διάγνωση CAD (Computer Aided

Diagnosis), η οποία σχετίζεται με την εφαρμογή τεχνικών επεξεργασίας και ανάλυσης

εικόνας αλλά και μηχανικής όρασης σε ψηφιοποιημένες μαστογραφικές εικόνες, και

(β) στη βελτιστοποίηση της ποιότητας εικόνας με ταυτόχρονη μείωση της δόσης στον

μαστό με τον σχεδιασμό και την εκλέπτυνση εξειδικευμένου μαστογραφικού εξοπλισμού

και τον προσδιορισμό των βέλτιστων λειτουργικών παραμέτρων ενός μαστογράφου.

Παρά το γεγονός ότι η έρευνα CAD παρουσιάζει εξαιρετική πρόοδο, η επιτυχία της

εξαρτάται από την ποιότητα της μαστογραφικής εικόνας η οποία λαμβάνεται στον

ανιχνευτή εικόνας.

Ο ανιχνευτής εικόνας είναι ένας από τους πιο καθοριστικούς παράγοντες της

αποτελεσματικότητας της μαστογραφικής διαδικασίας. Η μαστογραφία με συστήματα

φιλμ-ενισχυτικής πινακίδας παραμένει μέχρι και σήμερα η πιο διαδεδομένη τεχνική.

Μεταξύ άλλων, προσφέρει καλή απεικόνιση δομών χαμηλής αντίθεσης με εμφανή όρια.

Εντούτοις, τα συστήματα αυτά εχουν περιορισμένο εύρος έκθεσης (1:25) ενώ οι μάζες και

οι μικροασβεστώσεις, σημαντικές ενδείξεις ύπαρξης καρκίνου, δύσκολα απεικονίζονται

σε πυκνούς μαστούς.

Η πρόσφατη έρευνα έδειξε ότι η ψηφιακή μαστογραφία προσφέρει βελτιωμένη

ποιότητα εικόνας συγκριτικά με τα συστήματα φιλμ-ενισχυτικής πινακίδας καθώς επίσης

καλύτερη κβαντική αποδοτικότητα (quantum efficiency) και ευκολότερη λήψη,

επεξεργασία και αποθήκευση εικόνας. Επιπρόσθετα, οι φωτοαγώγιμοι ανιχνευτές ενεργού

151

ABSTRACT

μήτρας αποδεικνύονται να υπερέχουν των φωτοδιεγειρόμενων φωσφόρων

(photostimulable phosphors) και των συσκευών συζευγμένου φορτίου (Charge Coupled

Devices ή CCDs). Ειδικότερα, οι φωτοαγώγιμοι ανιχνευτές ενεργού μήτρας άμεσης

μετατροπής παρέχουν βελτιωμένη κβαντική αποδοτικότητα, μειωμένη ασάφεια και υψηλή

διακριτική ικανότητα.

Στους άμεσους ανιχνευτές ενεργού μήτρας, ένας φωτοαγωγός μετατρέπει άμεσα τις

προσπίπτουσες ακτίνες Χ σε νέφος φορτίων το οποίο ολισθαίνει κάτω από την επίδραση

ηλεκτρικού πεδίου προς τα ηλεκτρόδια όπου και συλλέγεται σχηματίζοντας την

μαστογραφική εικόνα. Ως εκ τούτου, το φωτοαγώγιμο υλικό είναι ένας από τους πιο

σημαντικούς παράγοντες σε αυτά τα συστήματα. Το άμορφο σελήνιο (a-Se) είναι ένα από

τα καταλληλότερα υλικά κυρίως λόγω της ικανότητάς του να αναπτύσσεται σε μεγάλες

επιφάνειες με ομοιογενή χαρακτηριστικά απεικόνισης και λόγω της υψηλής ενδογενούς

διακριτικής του ικανότητας. Παρόλαυτα, το υλικό αυτό πάσχει από περιορισμένη

ικανότητα απορρόφησης ακτίνων Χ και μειωμένη ευαισθησία. Υλικά όπως τα a-As2Se3,

GaSe, GaAs, Ge, CdTe, CdZnTe, Cd0.8Zn0.2Te, ZnTe, PbO, TlBr, PbI2 και HgI2

ικανοποιούν κάποια από τα χαρακτηριστικά ενός ιδανικού ανιχνευτή και έτσι είναι

υποψήφια σαν εναλλακτική λύση για αυτού του είδους τα συστήματα απεικόνισης.

Μεταξύ αυτών, τα πολυκρυσταλλικά CdTe, CdZnTe, Cd0.8Zn0.2Te, PbO, PbI2 και HgI2

όπως και το άμορφο a-As2Se3 είναι οι καλύτεροι δυνατοί υποψήφιοι κυρίως λόγω του

γεγονότος ότι μπορούν να αναπτυχθούν σε μεγάλες επιφάνειες. Απο την άλλη τα

κρυσταλλικά GaSe, GaAs, Ge, ZnTe και TlBr αναπτύσσονται σε περιορισμένες

επιφάνειες με βάση τις παρούσες τεχνικές και έτσι είναι λιγότερο κατάλληλα. Εντούτοις

οι τεχνικές ανάπτυξης βελτιώνονται.

Για να βελτιστοποιηθεί η ποιότητα της μαστογραφικής εικόνας και άρα η

διαγνωστική πληροφορία η οποία λαμβάνεται στους άμεσους ψηφιακούς ανιχνευτές,

απαιτείται μία προσεκτική επιλογή του φωτοαγώγιμου υλικού με μία ταυτόχρονη

εκλέπτυνση της τεχνολογίας του ανιχνευτή. Αυτά μπορούν να επιτευχθούν με την μελέτη

της φυσικής των διαδικασιών σχηματισμού του σήματος στα υλικά τα οποία

προαναφέρθηκαν αφού με αυτόν τον τρόπο λαμβάνονται σημαντικές πληροφορίες

σχετικά με το σχηματισμό της τελικής εικόνας καθώς επίσης και για τους παράγοντες που

δια μορφώνουν και επηρεάζουν την ποιότητά της.

Η ποιότητα της μαστογραφικής εικόνας σχετίζεται άμεσα με τα χαρακτηριστικά της.

Τα πρωτογενή ηλεκτρόνια τα οποία παράγονται εντός του φωτοαγώγιμου υλικού κατά

την ακτινοβόληση, αποτελούν το πρωτογενές σήμα το οποίο προχωρώντας σχηματίζει το

152

ABSTRACT

τελικό σήμα (εικόνα) στα ηλεκτρόδια του ανιχνευτή. Ως εκ τούτου, τα χαρακτηριστικά

της μαστογραφικής εικόνας εξαρτώνται άμεσα από τα χαρακτηριστικά των πρωτογενών

ηλεκτρονίων. Η πειραματική έρευνα δεν μπορεί να απομονώσει και να μελετήσει

αποκλειστικά και μόνο τα πρωτογενή ηλεκτρόνια. Από την άλλη, η έρευνα η οποία έχει

γίνει στο πεδίο της φυσικής του σχηματισμού του σήματος έχει ασχοληθεί κύρια με τα

φαινόμενα υστέρησης (lag και ghosting φαινόμενα), με την ευαισθησία και την

φωτοαγωγιμότητα όπως επίσης με την ολίσθηση, τον πολλαπλασιασμό, την

επανασύνδεση και την συλλογή των ηλεκτρικών φορέων. Παράλληλα, παρά το γεγονός

ότι υπάρχει ένας σημαντικός αριθμός εμπορικά διαθέσιμων πακέτων προσομοίωσης που

κάνουν χρήση των τεχνικών Monte Carlo, όπως είναι για παράδειγμα το EGS4 και το

PENELOPE, δεν έχουν πραγματοποιηθεί στα υλικά τα οποία προαναφέρθηκαν μελέτες

επί των χαρακτηριστικών των πρωτογενών ηλεκτρονίων όπως είναι ο αριθμός τους και οι

ενεργειακές, γωνιακές και χωρικές κατανομές τους ούτε έχει διερευνηθεί η επίδραση

αυτών των χαρακτηριστικών στην τελική εικόνα.

Στην παρούσα Διδακτορική Διατριβή διερευνόνται οι διαδικασίες σχηματισμού του

αρχικού σήματος και τα χαρακτηριστικά των πρωτογενών ηλεκτρονίων στα

φωτογαγώγιμα υλικά τα οποία προαναφέρθηκαν. Παράλληλα, πραγματοποιείται μία

πρώτη μελέτη της επίδρασης των χαρακτηριστικών των πρωτογενών ηλεκτρονίων στα

χαρακτηριστικά του τελικού σήματος, όπως και της κατανομής του ηλεκτρικού πεδίου και

των ηλεκτρονιακών αλληλεπιδράσεων ειδικότερα στο a-Se. Το ηλεκτρικό πεδίο και οι

ηλεκτρονιακές αλληλεπιδράσεις είναι δύο βασικές παράμετροι στην ανάπτυξη ενός

ολοκληρωμένου μοντέλου το οποίο θα προσομοιώνει το σχηματισμό του τελικού σήματος

και έτσι θα μελετά πλήρως την επίδραση των χαρακτηριστικών των πρωτογενών

ηλεκτρονίων στα χαρακτηριστικά της μαστογραφικής εικόνας.

Συγκεκριμένα, αναπτύσσεται ένα μοντέλο το οποίο χρησιμοποιεί τεχνικές Monte

Carlo για την προσομοίωση της παραγωγής των πρωτογενών ηλεκτρονίων κατά την

ακτινοβόληση στα φωτοαγώγιμα υλικά: a-Se, a-As2Se3, GaSe, GaAs, Ge, CdTe, CdZnTe,

Cd0.8Zn0.2Te, ZnTe, PbO, TlBr, PbI2 και HgI2. Το μοντέλο προσομοιώνει τις

αλληλεπιδράσεις των φωτονίων με το υλικό του ανιχνευτή (φωτοηλεκτρική απορρόφηση,

σκέδαση Compton και σκέδαση Rayleigh) όπως επίσης τις διαδικασίες ατομικής

αποδιέγερσης (εκπομπή φωτονίων φθορισμού, Auger και Coster-Kronig ηλεκτρονίων). Οι

παραδοχές οι οποίες γίνονται στο μοντέλο προκύπτουν από το συμβιβασμό μεταξύ

ακρίβειας και αλγοριθμικής απλότητας. Θεωρείται ότι σε όλα τα στοιχεία εκτός των

βαρεών Hg, Tl και Pb, η φωτοηλεκτρική απορρόφηση ενός φωτονίου που έχει ενέργεια

153

ABSTRACT

μεγαλύτερη από την ενέργεια σύνδεσης του Κ φλοιού, πραγματοποιείται αποκλειστικά

και μόνο από αυτόν τον φλοιό. Αυτό δίνει για προσπίπτον μονοενεργειακό φάσμα

ενέργειας 20 keV στο a-Se, 2.128 % σχετική διαφορά μεταξύ του αριθμού των

ηλεκτρονίων που υπολογίζεται με το συγκεκριμένο μοντέλο και του αριθμού που

υπολογίζεται με χρήση ενεργών διατομών για την φωτοηλεκτρική απορρόφηση από τους

διαφόρους φλοιούς. Η αντίστοιχη τιμή για τη συνολική ενέργεια των πρωτογενών

ηλεκτρονίων είναι 2.33 %. Επιπρόσθετα, η αποδιέγερση του Μ και των εξώτατων φλοιών

δε λαμβάνεται υπόψιν. Αυτό έχει σαν αποτέλεσμα να γίνεται μία υποεκτίμηση του

αριθμού των πρωτογενών ηλεκτρονίων με ενέργειες μικρότερες των 4 keV στα Hg, Tl και

Pb ιδιαίτερα για ενέργειες ακτίνων Χ μικρότερες από την ενέργεια σύνδεσης του LIII

υποφλοιού αυτών των στοιχείων. Παρόλαυτα, αφού η μέση μαστογραφική ενέργεια είναι

της τάξεως των 20 keV ενώ παράλληλα τα χαμηλής ενέργειας φωτόνια απορροφόνται

ισχυρά από το μαστό, αυτή η υποεκτίμηση δεν θεωρείται σημαντική συγκρινόμενη με την

προσπάθεια διατήρησης της αλγοριθμικής πολυπλοκότητας σε αποδεκτά όρια.

Τα αποτελέσματα τα οποία λαμβάνονται από την προσομοίωση χωρίζονται σε

τέσσερις κατηγορίες. Η πρώτη κατηγορία αφορά τις ενεργειακές κατανομές των

φθοριζόντων φωτονίων, των προσπιπτόντων και φθοριζόντων φωτονίων τα οποία

διαφεύγουν εμπρόσθια και οπίσθια καθώς επίσης των πρωτογενών ηλεκτρονίων. Η

δεύτερη κατηγορία σχετίζεται με τις αζιμουθιακές και πολικές γωνιακές κατανομές των

πρωτογενών ηλεκτρονίων. Η τρίτη κατηγορία αφορά τις χωρικές κατανομές των

πρωτογενών ηλεκτρονίων και τέλος η τέταρτη κατηγορία τις αριθμητικές κατανομές των

φθοριζόντων φωτονίων, των προσπιπτόντων και φθοριζόντων φωτονίων τα οποία

διαφεύγουν εμπρόσθια και οπίσθια καθώς επίσης των πρωτογενών ηλεκτρονίων. Τα

αποτελέσματα αυτά αφορούν 39 μονοενεργειακά φάσματα, με ενέργειες μεταξύ 2 και

40 keV, και 53 μαστογραφικά φάσματα στα οποία οι πλειονότητα των φωτονίων έχει

ενέργειες μεταξύ 15 και 40 keV. Τα φωτόνια Χ προσπίπτουν κάθετα στο κέντρο ενός

ανιχνευτή διαστάσεων 10 cm πλάτος, 10 cm μήκος και 1 mm πάχος. Η επιλογή του 1 mm

πάχους έγινε ώστε ο αριθμός τόσο των προσπιπτόντων όσο και των φθοριζόντων

φωτονίων τα οποία διαφεύγουν εμπρόσθια να είναι αμελητέος, αφού τα προσπίπτοντα και

τα φθορίζοντα φωτόνια είναι η κύρια πηγή των πρωτογενών ηλεκτρονίων από τη

φωτοηλεκτρική απορρόφησή τους.

Επιπρόσθετα, αναπτύσσεται ένας μαθηματικός φορμαλισμός για την ολίσθηση των

πρωτογενών ηλεκτρονίων του a-Se στο κενό κάτω από την επίδραση ενός ηλεκτρικού

πεδίου πυκνωτή και μελετώνται οι παραγόμενες ενεργειακές, γωνιακές και χωρικές

154

ABSTRACT

κατανομές των ηλεκτρονίων στο ηλεκτρόδιο συλλογής. Ο φορμαλισμός βασίζεται στις

εξισώσεις κίνησης του Νεύτωνα και στο θεώρημα μεταβολής της κινητικής ενέργειας.

Επίσης, ένα ήδη ανεπτυγμένο μοντέλο υπολογισμού του ηλεκτρικού πεδίου σε

άμεσους ανιχνευτές a-Se υιοθετείται και επανεξετάζεται ώστε να προσαρμοστεί στο

μοντέλο των πρωτογενών ηλεκτρονίων. Αναπτύσσεται κώδικας ο οποίος υπολογίζει την

κατανομή του ηλεκτρικού δυναμικού οπουδήποτε στο a-Se πάνω από τα pixel και τα κενά

μεταξύ των pixel της ενεργού μήτρας, χρησιμοποιώντας την υπάρχουσα αναλυτική λύση,

τις συνοριακές συνθήκες της περίπτωσής μας και την αριθμητική μέθοδο Gauss-Jordan

Elimination.

Τέλος, αναπτύσσεται η δομή και ο μαθηματικός φορμαλισμός ενός μοντέλου το

οποίο θα προσομοιώνει τις αλληλεπιδράσεις των ηλεκτρονίων στο a-Se, με βάση

υπάρχοντα μοντέλα προσομοίωσης και ανεπτυγμένες θεωρείες της φυσικής των

ηλεκτρονιακών αλληλεπιδράσεων. Ο φορμαλισμός περιλαμβάνει την ελεύθερη διαδρομή

των ηλεκτρονίων, τη διαδικασία επιλογής του είδους της ηλεκτρονιακής αλληλεπίδρασης,

τη διαφορική και ολική ενεργό διατομή της ελαστικής σκέδασης και τη διαφορική και

ολική ενεργό διατομή της ανελαστικής σκέδασης τόσο με τους εσωτερικούς φλοιούς

(Κ και L φλοιοί) όσο και με τους εξωτερικούς.

Τα αποτελέσματα δείχνουν ότι οι ενεργειακές κατανομές των προσπιπτόντων

φωτονίων τα οποία διαφεύγουν οπίσθια έχουν παρόμοια μορφή με το φάσμα ακτίνων Χ

ενώ δεν ισχύει το ίδιο για τις κατανομές των φωτονίων τα οποία διαφεύγουν εμπρόσθια.

Τα προσπίπτοντα φωτόνια που διαφεύγουν εμπρόσθια έχουν υψηλές ενέργειες και αρκετά

πάνω από τις αιχμές απορρόφησης.

Επίσης, το χαρακτηριστικό γνώρισμα στις ενεργειακές κατανομές των πρωτογενών

ηλεκτρονίων των PbI2 και HgI2 είναι οι αιχμές των ατομικών αποδιεγέρσεων. Λόγω του

γεγονότος ότι η φωτοηλεκτρική απορρόφηση είναι η κύρια μορφή αλληλεπίδρασης των

ακτίνων Χ με την ύλη στις μαστογραφικές ενέργειες, τα πρωτογενή ηλεκτρόνια

αποτελούνται από φωτοηλεκτρόνια, Auger και Coster-Kronig ηλεκτρόνια. Ως εκ τούτου,

οι αιχμές αποδιέγερσης αποτελούνται από φωτοηλεκτρόνια τα οποία προέρχονται από την

απορρόφηση των φωτονίων φθορισμού όπως επίσης από Auger και Coster-Kronig

ηλεκτρόνια. Στα υπόλοιπα υλικά τα φωτοηλεκτρόνια τα οποία παράγονται από την

απορρόφηση των προσπιπτόντων φωτονίων επηρεάζουν επιπρόσθετα τη μορφή των

κατανομών. Ειδικότερα, προσδίδουν στις κατανομές μία μορφή παρόμοια με εκείνη του

προσπίπτοντος φάσματος μετατοπισμένο όμως σε χαμηλότερες ενέργειες.

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ABSTRACT

Τα πρωτογενή ηλεκτρόνια προτιμούν να εκπέμπονται εμπρόσθια. Στις

μαστογραφικές ενέργειες, το ποσοστό των ηλεκτρονίων που εκπέμπονται εμπρόσθια είναι

της τάξεως του 60 % με την πιο πιθανή πολική γωνία εκπομπής να κυμαίνεται από 50ο

μέχρι 70ο. Επιπρόσθετα, τα ηλεκτρόνια προτιμούν να εκπέμπονται σε δύο λοβούς γύρω

από τις αζιμουθιακές γωνίες φ=0 και φ=π. Αντιθέτως, έχουν την μικρότερη πιθανότητα

εκπομπής στις αζιμουθιακές γωνίες φ= π/2 και 3π/2 και παράλληλα με τον άξονα της

δέσμης, τόσο εμπρόσθια όσο και οπίσθια. Η παρουσία ηλεκτρονίων Auger και Coster-

Kronig αυξάνει την αζιμουθιακή ομοιομορφία, κάτι το οποίο σημαίνει μικρότερη τάση

εκπομπής ηλεκτρονίων στις φ=0, π και 2π. Αυτό οφείλεται στο γεγονός ότι τα ηλεκτρόνια

αυτά εκπέμονται ισοτροπικά στο χώρο. Στις πρακτικές μαστογραφικές ενέργειες

(15-40 keV) τα a-Se, a-As2Se3 και Ge έχουν την ελάχιστη αζιμουθιακή ομοιομορφία ενώ

τα CdZnTe, Cd0.8Zn0.2Te και CdTe τη μέγιστη. Η αζιμουθιακή ομοιομορφία είναι μία από

τις παραμέτρους οι οποίες καθορίζουν τις τροχιές ολίσθησης των ηλεκτρονίων κατά την

εφαρμογή ενός ηλεκτρικού πεδίου και ως εκ τούτου είναι ένας παράγοντας ο οποίος

επηρεάζει τα χαρακτηριστικά της τελικής εικόνας.

Κατά προσέγγιση το 80 % των πρωτογενών ηλεκτρονίων παράγονται στο σημείο

πρόσπτωσης των ακτίνων Χ σε όλα τα υλικά τα οποία μελετώνται. Το γεγονός αυτό

οφείλεται τόσο στο ότι η φωτοηλεκτρική απορρόφηση των προσπιπτόντων φωτονίων, η

οποία ακολουθείται από την ατομική αποδιέγερση κατά τη διάρκεια της οποίας

παράγονται Auger και Coster Kronig ηλεκτρόνια, λαμβάνει χώρα σχεδόν αποκλειστικά

στο σημείο αυτό όσο και στο ότι τα προσπίπτοντα φωτόνια τα οποία σκεδάζονται με

Compton παράγουν και αυτά με τη σειρά τους πρωτογενή ηλεκτρόνια στο ίδιο σημείο.

Τόσο οι xy (στο επίπεδο του ανιχνευτή) όσο και οι yz (κατά βάθος) ηλεκτρονιακές

χωρικές κατανομές επηρεάζονται από τη σκέδαση των φωτονίων όπως επίσης από την

εκπομπή φωτονίων φθορισμού. Οι κατανομές των a-Se, a-As2Se3, GaSe, GaAs, Ge, PbO

και TlBr είναι σχεδόν ανεξάρτητες του μαστογραφικού φάσματος, μιας και οι αιχμές

απορρόφησής τους έχουν χαμηλές ενέργειες, ενώ αυτές των CdTe, CdZnTe, Cd0.8Zn0.2Te,

ZnTe, PbI2 και HgI2 παρουσιάζουν φασματική εξάρτηση, λόγω του ότι κάποιες αιχμές

απορρόφησής τους έχουν υψηλότερες ενέργειες. Για το πρακτικό μαστογραφικό εύρος

ενεργειών και στο πρωταρχικό αυτό στάδιο της δημιουργίας του πρωτογενούς σήματος,

το a-Se παρουσιάζει την καλύτερη ενδογενή χωρική διακριτική ικανότητα συγκριτικά με

τα υπόλοιπα υλικά. Το γεγονός αυτό μπορεί να αποτελεί μία ένδειξη ότι η διακριτική

ικανότητα του a-Se είναι ανώτερη. Για όλα τα υλικά τα όποια μελετώνται καθώς επίσης

για όλα τα προσπίπτοντα φάσματα ακτίνων Χ, η πλειονότητα των πρωτογενών

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ABSTRACT

ηλεκτρονίων παράγεται εντός των πρώτων 300 μm από την επιφάνεια του ανιχνευτή. Το

PbO παρουσιάζει το μικρότερο χώρο μέσα στον οποίο μπορούν να παραχθούν πρωτογενή

ηλεκτρόνια, με ακτίνα R=200 μm και βάθος Dmax=320 μm, ενώ το CdTe το μεγαλύτερο,

με ακτίνα R=500 μm και βάθος Dmax=660 μm.

Στο στάδιο παραγωγής του αρχικού σήματος και για τα τυπικά πάχη ανιχνευτών

(300-1000 μm), το μέσο ποσοστό της προσπίπτουσας ενέργειας το οποίο μεταφέρεται στα

πρωτογενή ηλεκτρόνια είναι 97 % ενώ το μικρότερο είναι 84.5 % (CdTe στα 32 keV). Το

μέγιστο ποσοστό διαφυγέντων φωτονίων φθορισμού είναι 30.701 % (a-Se στα 13 keV)

ενώ ο μέσος όρος είναι 7.482 %. Οι αντίστοιχες τιμές για τα διαφυγέντα προσπίπτοντα

φωτόνια είναι 6 % (GaSe στα 40 keV) και 0.405%. Σε όλα τα υλικά και τις ενέργειες,

εκτός των E 30 keV στα a-Se, a-As2Se3, GaSe, GaAs και Ge (ελαφρά υλικά), τα φωτόνια

διαφεύγουν οπίσθια ενώ η συντριπτική τους πλειονότητα είναι φωτόνια φθορισμού. Η

διαφυγή των φωτονίων φθορισμού και η ατομική αποδιέγερση είναι οι παράγοντες οι

οποίοι επηρεάζουν την παραγωγή των πρωτογενών ηλεκτρονίων. Ο αριθμός τους αυξάνει

σε ενέργειες μεγαλύτερες των Κ αιχμών των ελαφρών υλικών, των Κ αιχμών του Cd και

του Τe όπως και των L αιχμών των Pb, Hg και Τl στις οποίες παρατηρείται μείωση της

διαφυγής των φθοριζόντων φωτονίων και η απορρόφησή τους συνοδεύεται από

μακροσκελείς ατομικές αποδιεγέρσεις. Για ενέργειες Ε 30 keV στα ελαφρά υλικά, ο

αριθμός των φωτονίων τα οποία διαφεύγουν εμπρόσθια αυξάνει, λόγω της διαφυγής

προσπιπτόντων φωτονίων, και γίνεται μεγαλύτερος του αριθμού των φωτονίων που

διαφεύγουν οπίσθια. Επιπρόσθετα, η παραγωγή πρωτογενών ηλεκτρονίων επηρεάζεται

και από τη διαφυγή των πρωτογενών φωτονίων η οποία μειώνει τον αριθμό τους. Το a-Se

έχει τον μικρότερο παραγόμενο αριθμό πρωτογενών ηλεκτρονίων για το πρακτικό

μαστογραφικό εύρος ενεργειών.

Τα αποτελέσματα τα οποία αφορούν τα πρωτογενή ηλεκτρόνια του a-Se τα οποία

ολισθαίνουν στο κενό κάτω από την επίδραση ηλεκτρικού πεδίου πυκνωτή και φτάνουν

στο ηλεκτρόδιο συλλογής, δίνουν σε πρώτη προσέγγιση την επίδραση των

χαρακτηριστικών του πρωτογενούς σήματος στα χαρακτηριστικά του τελικού σήματος.

Οι ενεργειακές κατανομές των ηλεκτρονίων είναι μετατοπισμένες σε λίγο μεγαλύτερες

ενέργειες ενώ υπάρχει και μία μικρή διαφοροποίηση της μορφής τους. Αυτό οφείλεται

στο γεγονός ότι, όπως σημειώθηκε πιο πάνω, η πλειονότητα των πρωτογενών

ηλεκτρονίων παράγεται εντός των πρώτων 300 μm από την επιφάνεια του ανιχνευτή,

δηλαδή πολύ κοντά στο ηλεκτρόδιο συλλογής. Άμεσο επακόλουθο αυτού είναι ότι τα

περισσότερα ηλεκτρόνια σαρώνονται σε χρόνους t<5 x 10-12 s ενώ το σήμα (ηλεκτρικός

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ABSTRACT

παλμός) έχει μία διάρκεια μικρότερη των 7.2 x 10-11 s. Λόγω του ότι το 80 % των

πρωτογενών ηλεκτρονίων παράγεται στο σημείο πρόσπτωσης των ακτίνων Χ, η

πλειονότητα των ηλεκτρονίων συλλέγεται στο σημείο αυτό. Το FWHM της PSF των

πρωτογενών ηλεκτρονίων στο ηλεκτρόδιο υψηλής τάσης είναι σχεδόν 5.5 φορές

μεγαλύτερο από την αρχική του τιμή. Επιπρόσθετα, οι xy χωρικές κατανομές τους

παρουσιάζουν δύο αντιδιαμετρικά αντίθετους λοβούς γύρω από τον άξονα y=0 όπως

επίσης έναν δακτύλιο ακτίνας 2 mm περίπου. Οι δύο λοβοί οφείλονται στο γεγονός ότι το

ηλεκτρικό πεδίο εφαρμόζεται κάθετα στον ανιχνευτή με αποτέλεσμα οι αζιμουθιακές

γωνιακές κατανομές των ηλεκτρονίων να μην επηρεάζονται κατά την ολίσθησή τους. Ο

δακτύλιος οφείλεται στα ηλεκτρόνια Auger τα οποία εκπέμπονται ισοτροπικά. Τέλος,

όλα τα συλλεγόμενα ηλεκτρόνια έχουν πολικές γωνίες θ>π/2 με την πιο πιθανή γωνία να

είναι η θ=1.92 rad ~ 111o.

Συμπερασματικά, πραγματοποιείται καταρχήν μία συγκριτική μελέτη μεταξύ των

διαφόρων φωτοαγώγιμων υλικών, σε μία πλειάδα μονονεργειακών και πολυενεργειακών

φασμάτων τα οποία καλύπτουν τις μαστογραφικές ενέργειες, και η οποία αφορά τον

αριθμό και την ενέργεια των φθοριζόντων και διαφυγέντων φωτονίων αλλά και τον

αριθμό, την ενέργεια και τις γωνιακές και χωρικές κατανομές των ηλεκτρονίων, στο

στάδιο σχηματισμού του πρωτογενούς σήματος. Η μονοενεργειακή περίπτωση, φωτίζει

τις φυσικές διαδικασίες που λαμβάνουν χώρα στην παραγωγή του αρχικού σήματος με

αποτέλεσμα να επιτρέπεται τόσο η διερεύνηση των χαρακτηριστικών των πρωτογενών

ηλεκτρονίων καθώς επίσης και των παραγόντων που επηρεάζουν αυτά τα

χαρακτηριστικά. Η πολυενεργειακή περίπτωση, παρέχει πληροφορία σε σχέση με την

εξάρτηση αυτών των χαρακτηριστικών από το προσπίπτον μαστογραφικό φάσμα. Έτσι

επιτρέπεται μία πρώτη επιλογή των πιο κατάλληλων υλικών. Λόγω του ότι τα TlBr,

GaAs, GaSe, ZnTe και CdTe παρουσιάζουν από τους μεγαλύτερους αριθμούς

πρωτογενών ηλεκτρονίων και έχουν υψηλή ευαισθησία (W ~6 eV), θα μπορούσαν να

αποτελέσουν την καλύτερη επιλογή με βάση το κριτήριο του υψηλού gain-amplification

του σήματος. Από την άλλη, τα a-Se, a-As2Se3 και Ge παρουσιάζουν από τις καλύτερες

ενδογενείς χωρικές διακριτικές ικανότητες και την μικρότερη αζιμουθιακή ομοιομορφία.

Δεδομένου ότι παρουσία ηλεκτρικού πεδίου μικρή αζιμουθιακή ομοιομορφία

συνεπάγεται υποβάθμιση της διακριτικής ικανότητας κυρίως σε μία διάσταση, αυτά τα

υλικά θα μπορούσαν να θεωρηθούν από τις καλύτερες επιλογές με βάση το κριτήριο της

υψηλής διακριτικής ικανότητας. Λόγω του ότι το PbO παρουσιάζει το μικρότερο βάθος

παραγωγής πρωτογενών ηλεκτρονίων (Dmax=320 μm), είναι η καλύτερη επιλογή με βάση

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ABSTRACT

το κριτήριο του ελάχιστου απαιτούμενου πάχους φωτοαγωγού. Τέλος τα PbI2 και HgI2 θα

μπορούσαν να αποτελέσουν τις καλύτερες επιλογές με βάση όλα τα κριτήρια αφού:

i. Έχουν υψηλή ευαισθησία (W ~4.5 eV) και έναν μέσο αριθμό πρωτογενών

ηλεκτρονίων.

ii. Η αζιμουθιακή τους ομοιομορφία, η διακριτική τους ικανότητα και το ελάχιστο

απαιτούμενο πάχος φωτοαγωγού κυμαίνονται στο μέσο όρο.

Σε αντίθεση με τα διαθέσιμα πακέτα προσομοίωσης, όπως το EGS4 και το

PENELOPE, στα οποία γίνεται χρήση ενεργών διατομών για την φωτοηλεκτρική

απορρόφηση από τους διαφόρους φλοιούς ενώ η ατομική αποδιέγερση ακολουθείται

μέχρι τους εξώτατους φλοιούς, οι προσεγγίσεις οι οποίες γίνονται στο παρόν μοντέλο

απλουστεύουν την φωτοηλεκτρική απορρόφηση και την ατομική αποδιέγερση. Έτσι, ο

υπολογιστικός χρόνος συντηρείται κάτω από αποδεκτά όρια (<20 min. για Pentium 4, 2.8

GHz, 448 MB RAM) ενώ ταυτόχρονα η εγκυρότητα και ακρίβεια των αποτελεσμάτων δεν

αλλοιώνεται.

Από την άλλη, τα αποτελέσματα τα οποία αφορούν τα πρωτογενή ηλεκτρόνια του

a-Se τα οποία ολισθαίνουν στο κενό κάτω από την επίδραση ηλεκτρικού πεδίου πυκνωτή

και φτάνουν στο ηλεκτρόδιο συλλογής, παρά το γεγονός ότι αφορούν μία μη ρεαλιστική

περίπτωση, εντούτοις δίνουν μία πρώτη προσέγγιση της επίδρασης των χαρακτηριστικών

των πρωτογενών ηλεκτρονίων στα χαρακτηριστικά του τελικού σήματος. Παρόλαυτα, μία

ολοκληρωτική προσομοίωση της διέλευσης του σήματος μέσα στον φωτοαγωγό μέχρι τα

ηλεκτρόδια συλλογής απαιτείται για να εξαχθούν τελικά συμπεράσματα που θα αφορούν

τη συσχέτιση των χαρακτηριστικών τελικού και πρωτογενούς σήματος κάτι το οποίο θα

επέτρεπε την επιλογή των πιο κατάλληλων φωτοαγώγιμων υλικών για τους άμεσους

ανιχνευτές ενεργού μήτρας και τη βελτιστοποίηση του σχεδιασμού τους. Τη βάση για την

ανάπτυξη ενός τέτοιου μοντέλου μπορούν να αποτελέσουν οι φορμαλισμοί που

παρουσιάζονται για τον υπολογισμό του ηλεκτρικού πεδίου και την προσομοίωση των

ηλεκτρονιακών αλληλεπιδράσεων στο a-Se. Εντούτοις, ο φορμαλισμός των

ηλεκτρονιακών αλληλεπιδράσεων στο a-Se χρειάζεται να εκλεπτυνθεί σε σχέση με την

φυσική του και να εμπλουτιστεί με ευκινησίες και χρόνους ζωής φορέων όπως επίσης με

μηχανισμούς παγίδευσης, επανασύνδεσης και ολίσθησης φορέων. Τέλος το μοντέλο θα

πρέπει να επιβεβαιωθεί είτε πειραματικά είτε θεωρητικά. Τα συγκεκριμένα ζητήματα θα

αποτελέσουν και το αντικείμενο της μελλοντικής έρευνας.

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