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ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΑΤΡΩΝ
ΤΜΗΜΑ ΙΑΤΡΙΚΗΣ
ΤΜΗΜΑ ΦΥΣΙΚΗΣ
ΔΙΑΤΜΗΜΑΤΙΚΟ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ
ΣΤΗΝ ΙΑΤΡΙΚΗ ΦΥΣΙΚΗ
ΑΝΑΠΤΥΞΗ ΜΕΘΟΔΩΝ ΠΡΟΣΟΜΟΙΩΣΗΣ
ΜΕ ΤΕΧΝΙΚΕΣ MONTE CARLO
ΔΙΑΔΙΚΑΣΙΩΝ ΠΑΡΑΓΩΓΗΣ ΣΗΜΑΤΟΣ ΣΕ ΦΩΤΟΑΓΩΓΙΜΑ
ΥΛΙΚΑ ΑΜΕΣΩΝ ΑΝΙΧΝΕΥΤΩΝ ΕΝΕΡΓΟΥ ΜΗΤΡΑΣ
ΣΤΗΝ ΨΗΦΙΑΚΗ ΜΑΣΤΟΓΡΑΦΙΑ
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
i
TABLE OF CONTENTS
Chapter 1. Introduction ............................................................................................................ 1
1.1. Introduction ....................................................................................................................... 2
1.2. Thesis ................................................................................................................................ 4
1.3. Thesis layout ..................................................................................................................... 6
1.4. Publications ....................................................................................................................... 6
1.5. Financial support ............................................................................................................... 7
Chapter 2. Mammography........................................................................................................ 8
2.1. Introduction ....................................................................................................................... 9
2.2. Mammographic Equipment .............................................................................................. 9
2.2.1. X-ray tube ................................................................................................................ 10
2.2.2. Compression device ................................................................................................. 11
2.2.3. Antiscatter grid......................................................................................................... 11
2.2.4. Image receptors ........................................................................................................ 12
Chapter 3. Mammographic Detectors ................................................................................... 13
3.1. Introduction ..................................................................................................................... 14
3.2. Phosphors ........................................................................................................................ 14
3.3. Film-Screen systems (Analog detectors) ........................................................................ 14
3.4. Digital Detectors ............................................................................................................. 15
3.4.1. Phosphor-CCD systems ........................................................................................... 15
3.4.1.1. CCD devices...................................................................................................... 15
3.4.1.2. Optical coupling of a phosphor to a CCD ........................................................ 16
3.4.1.3. Slot scanned digital mammography .................................................................. 18
3.4.2. Photostimulable phosphors (Computed radiography systems) ................................ 18
3.4.3. Active matrix flat panel imagers (AMFPI) .............................................................. 20
3.4.3.1. Indirect detection .............................................................................................. 20
3.4.3.2. Direct detection ................................................................................................. 21
3.5. Digital versus Film-Screen Mammography .................................................................... 24
Chapter 4. X-Ray Photoconductors ....................................................................................... 26
4.1. Introduction ..................................................................................................................... 27
4.2. The ideal x-ray photoconductor ...................................................................................... 27
4.3. Crystalline, amorphous and polycrystalline solids ......................................................... 28
4.4. X-ray photoconductors.................................................................................................... 29
4.4.1. Amorphous Selenium (a-Se) .................................................................................... 30
4.4.2. Polycrystalline Cadmium Telluride (poly-CdTe) .................................................... 31
4.4.3. Polycrystalline Cadmium Zinc Telluride (poly-CdZnTe) ....................................... 31
4.4.4. Polycrystalline Lead Oxide (poly-PbO)................................................................... 32
4.4.5. Polycrystalline Mercuric Iodide (poly-HgI2) ........................................................... 32
4.4.6. Polycrystalline Lead Iodide (poly-PbI2) .................................................................. 33
4.5. Table of material properties ............................................................................................ 34
Chapter 5. Physics Of Image Formation ............................................................................... 35
5.1. Introduction ..................................................................................................................... 36
5.2. X-ray – matter interactions ............................................................................................. 36
5.2.1. Coherent (Rayleigh) scattering ................................................................................ 36
5.2.2. Incoherent (Compton) scattering ............................................................................. 37
5.2.3. Photoelectric absorption........................................................................................... 37
TABLE OF CONTENTS
ii
5.3. Atomic deexcitation ........................................................................................................ 38
5.4. Electron interactions ....................................................................................................... 40
5.4.1. Elastic scattering ...................................................................................................... 40
5.4.2. Inelastic scattering ................................................................................................... 42
5.4.2.1. The development of a theory for inelastic collisions ........................................ 42
5.4.2.2. Recoil energy .................................................................................................... 43
5.4.2.3. Bethe’s theory revisited..................................................................................... 44
5.4.2.4. Generalized Oscillator Strength -Optical Oscillator Strength ......................... 45
5.4.2.5. Bethe surface- Bethe sum rule .......................................................................... 47
5.4.2.6. The differential inelastic scattering cross section ............................................ 47
5.4.2.7. Secondary electron emission (δ-rays) ............................................................... 49
5.4.3. Collective description of electron interactions ........................................................ 49
5.5. Charge carrier transport inside a-Se ................................................................................ 50
5.5.1. Geminate (Onsager) recombination ......................................................................... 52
5.5.2. Columnar recombination ......................................................................................... 53
5.5.2.1. Absence of electric field .................................................................................... 54
5.5.2.2. Field parallel to the column .............................................................................. 55
5.5.2.3. Field perpendicular to the column .................................................................... 55
5.5.2.4. Field at an angle φ with the track ..................................................................... 55
Chapter 6. Monte Carlo Simulation ....................................................................................... 56
6.1. Introduction ..................................................................................................................... 57
6.2. Random numbers-Random variables .............................................................................. 58
6.3. Probability Distribution Functions - Cumulative Distribution Functions....................... 59
6.4. Sampling techniques ....................................................................................................... 60
6.4.1. The Inversion Method .............................................................................................. 61
6.4.2. The Rejection Method.............................................................................................. 62
Chapter 7. Primary Electron Generation Model .................................................................. 63
7.1. Introduction ..................................................................................................................... 64
7.2. Electron from Incoherent Scattering ............................................................................... 64
7.3. Photoelectric absorption.................................................................................................. 65
7.3.1. Photoelectric absorption from a molecule ............................................................... 65
7.3.2. Production of photoelectron ..................................................................................... 66
7.4. Atomic deexcitation ........................................................................................................ 68
7.4.1. K and L shell deexcitation ....................................................................................... 68
7.4.2. Simulated atomic transitions .................................................................................... 69
7.4.3. Energies and directions of fluorescent photons, Auger and CK electrons .............. 71
7.5. Model limitations ............................................................................................................ 71
Chapter 8. Primary Electron Generation: Results & Discussion ........................................ 73
8.1. Introduction ..................................................................................................................... 74
8.2. Energy distributions. ....................................................................................................... 75
8.2.1. Fluorescent photons. ................................................................................................ 75
8.2.2. Escaping photons. .................................................................................................... 75
8.2.3. Primary electrons. .................................................................................................... 75
8.2.3.1. Monoenergetic case. ......................................................................................... 77
8.2.3.2. Polyenergetic case. ........................................................................................... 78
8.3. Angular distributions of primary electrons. .................................................................... 80
8.3.1. Azimuthal distributions............................................................................................ 80
TABLE OF CONTENTS
iii
8.3.2. Polar distributions. ................................................................................................... 83
8.4. Spatial distributions of primary electrons. ...................................................................... 85
8.4.1. Monoenergetic case. ................................................................................................ 86
8.4.2. Polyenergetic case. ................................................................................................... 94
8.5. Arithmetics of photons and primary electrons. ............................................................... 95
8.5.1. Arithmetics of escaping primary photons. ............................................................... 96
8.5.2. Arithmetics of fluorescent photons produced. ......................................................... 96
8.5.3. Arithmetics of escaping fluorescent photons. .......................................................... 98
8.5.4. Arithmetics of escaping primary and fluorescent photons. ..................................... 98
8.5.5. Arithmetics of primary electrons produced. .......................................................... 100
8.5.6. Summary tables. ..................................................................................................... 101
Chapter 9. A Preliminary Study On Final Signal Formation In a-Se .............................. 103
9.1. Introduction ................................................................................................................... 104
9.2. Mathematical formulation ............................................................................................. 104
9.3. Results and Discussion ................................................................................................. 106
9.3.1. Energy distribution of primary electrons on top electrode .................................... 106
9.3.2. Time distribution of primary electrons on top electrode ....................................... 106
9.3.3. Spatial distribution of primary electrons on top electrode ..................................... 107
9.3.4. Angular distributions of primary electrons on top electrode ................. …………109
Chapter 10. Electric Field Considerations In a-Se ............................................................. 110
10.1. Introduction ................................................................................................................. 111
10.2. Boundary conditions ................................................................................................... 111
10.3. Calculation of the electric potential distribution ......................................................... 113
Chapter 11. Model Formulation For Electron Interactions In a-Se ................................. 117
11.1. Introduction ................................................................................................................. 118
11.2. Electron free path length ............................................................................................. 118
11.3. Decision on the type of electron interaction ............................................................... 119
11.4. Elastic scattering ......................................................................................................... 119
11.4.1. Differential cross section ..................................................................................... 119
11.4.2. Elastic scattering cross section (σel) ..................................................................... 121
11.5. Inelastic scattering ...................................................................................................... 122
11.5.1. Inelastic scattering with inner shells (K and L shells) ......................................... 122
11.5.2. Inelastic scattering with outer shells .................................................................... 124
Chapter 12. General Discussion, Conclusions & Future Work ......................................... 128
12.1. General discussion ...................................................................................................... 129
12.2. Conclusions and future work ...................................................................................... 133
References ............................................................................................................................... 136
Abstract................................................................................................................................... 144
Εκτενής Περίληψη ................................................................................................................. 147
CHAPTER 1: INTRODUCTION
CHAPTER 1
INTRODUCTION
1
CHAPTER 1: INTRODUCTION
2
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
CHAPTER 1: INTRODUCTION
3
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,
CHAPTER 1: INTRODUCTION
4
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.
CHAPTER 1: INTRODUCTION
5
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
CHAPTER 1: INTRODUCTION
6
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.
CHAPTER 1: INTRODUCTION
7
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).
CHAPTER 2: MAMMOGRAPHY
CHAPTER 2
MAMMOGRAPHY
8
CHAPTER 2: MAMMOGRAPHY
9
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).
X-ray tube
Compression
device
Image
receptor
Figure 2.1. A typical mammographic unit.
CHAPTER 2: MAMMOGRAPHY
10
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
same element as the target) selectively attenuates and optimizes the beam spectrum.
Collimation of the x-ray beam is accomplished usually by diaphragms. Diaphragm
CHAPTER 2: MAMMOGRAPHY
11
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.
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
CHAPTER 2: MAMMOGRAPHY
12
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.
CHAPTER 3: MAMMOGRAPHIC DETECTORS
CHAPTER 3
MAMMOGRAPHIC DETECTORS
13
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
CHAPTER 3: MAMMOGRAPHIC DETECTORS
15
Incident X-rays
Film
Phosphor screen
Visible light
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
CHAPTER 3: MAMMOGRAPHIC DETECTORS
16
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
CHAPTER 3: MAMMOGRAPHIC DETECTORS
17
(a)
(b)
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.
Phosphor
Demagnifying
fibre optic taper
CCD
Figure 3.4. A full-field digital breast imaging system composed of a matrix of phosphors
coupled to CCDs by fibre optics.
CHAPTER 3: MAMMOGRAPHIC DETECTORS
18
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
CHAPTER 3: MAMMOGRAPHIC DETECTORS
Figure 3.6.
19
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.
CHAPTER 3: MAMMOGRAPHIC DETECTORS
20
Scattering of the light within the phosphor causes release of traps over a 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.
(a)
(b)
Figure 3.7. (a) A cross section and (b) a microphotograph of a pixel for an indirect
AMFPI (Antonuk et al 2000).
CHAPTER 3: MAMMOGRAPHIC DETECTORS
21
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
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).
CHAPTER 3: MAMMOGRAPHIC DETECTORS
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
CHAPTER 3: MAMMOGRAPHIC DETECTORS
23
acoll
Pixel width (a)
(a)
(b)
Pixel pitch (d)
(c)
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
CHAPTER 3: MAMMOGRAPHIC DETECTORS
24
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).
Detector parameter
Value
Detector size (cm2)
Pixel pitch (μm)
Number of pixels
Readout time (s)
X-ray spectrum (kVp)
Mean exposure (mR)
Exposure range (mR)
18 x 24
50
3600 x 4800
<5
30
12
0.6-240
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
CHAPTER 3: MAMMOGRAPHIC DETECTORS
25
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.
CHAPTER 4: X-RAY PHOTOCONDUCTORS
CHAPTER 4
X-RAY PHOTOCONDUCTORS
26
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 ( W ) 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.
CHAPTER 4: X-RAY PHOTOCONDUCTORS
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 W 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).
CHAPTER 4: X-RAY PHOTOCONDUCTORS
(a)
29
(b)
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).
CHAPTER 4: X-RAY PHOTOCONDUCTORS
30
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
CHAPTER 4: X-RAY PHOTOCONDUCTORS
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 ( W ). 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 W is about 45 eV when for polycrystalline mercuric iodide (poly-HgI2) and
polycrystalline Cadmium Zinc Telluride (poly-CdZnTe) the value of W is typically
5-6 eV.
4.4.2. Polycrystalline Cadmium Telluride (poly-CdTe)
CdTe has a moderate atomic number (Zeff ~ 50) and a low W ~ 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
CHAPTER 4: X-RAY PHOTOCONDUCTORS
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 W ~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
CHAPTER 4: X-RAY PHOTOCONDUCTORS
33
poly-HgI2 layer, respectively (Street et al 2002, Zentai et al 2004). HgI2 has W ~ 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 W ~ 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
CHAPTER 4: X-RAY PHOTOCONDUCTORS
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).
Photoconductor
/state
a-Se
Amorphous
a-As2Se3
Amorphous
GaSe
Crystalline
GaAs
Crystalline
Ge
Crystalline
CdTe
Polycrystalline
CdZnTe,
Cd0.8Zn0.2Te
Polycrystalline
ZnTe
Crystalline
PbO
Polycrystalline
TlBr
Crystalline
PbI2
Polycrystalline
HgI2
Polycrystalline
Eg
(eV)
W
(eV)
45
(at 10 V/μm)
Density
(g/cm3)
Resistivity
(Ω cm)
4.3
1014-1015
1.8
~ 4.46
4.55
1012
2
6.3
4.6
1.42
6.3
5.31
107
0.7
1.5
5.32
102
1.5
4.65
6.06
109
1.7
5
5.8
1011
2.26
7
6.34
2.7
8-20
9.8
7-10 x 1012
2.7
6.5
7.5
1012
2.3
5
6.1
1011-1012
2.1
4.1
6.3
4 x 1013
2.3
CHAPTER 5: PHYSICS OF IMAGE FORMATION
CHAPTER 5
PHYSICS OF IMAGE FORMATION
35
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:
d o ro2
 (1  cos 2  )
d
2
(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:
CHAPTER 5: PHYSICS OF IMAGE FORMATION
37
d coh ro2
 (1  cos 2  )[ F ( ~
p, Z )] 2
d
2
(5.2)
where F( ~p , 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 ~p is related to the
momentum transfer during the collision given by:
 E 
~
p  29.1433 2  1  cos 
 mc 
(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 E  given by:
E 
E
E
1
(1  cos  )
mc 2
 E
(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( ~p , 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:
d incoh 1 2 2 1
 ro  (    sin 2  )S ( ~
p, Z )
d
2

(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:

E   
 tan  
cot  e  1 
2 
 mc   2 
(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
CHAPTER 5: PHYSICS OF IMAGE FORMATION
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):
dI
sin 2 
 const  cos 2 
4
d
E


  cos  
1 
2
 2mc

(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 Efl = 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 XiXjY CK in figure
5.1(c)). As it is the case for the fluorescent photons, the Auger and the CK electrons are
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.
CHAPTER 5: PHYSICS OF IMAGE FORMATION
39
Y-shell
Xi Y -photon
Xj-subshell
Xi Xj -photon
X-shell
Xi-subshell
(a)
XYZ-Auger
Z-shell
XYY-Auger
Y-shell
X-shell
(b)
Xi Xj Z-CK
Z-shell
Xi Xj Y-CK
Y-shell
Xj-subshell
X-shell
Xi-subshell
(c)
Figure 5.1. The three types of atomic deexcitation. (a) Fluorescent photon emission,
(b) Auger electron ejection and (c) CK electron ejection.
CHAPTER 5: PHYSICS OF IMAGE FORMATION
40
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):
1 r
 (r )   nuc (r )  4e


2
  (r )r  dr     (r )r dr 
r0
r





electron cloud
(5.10a)
CHAPTER 5: PHYSICS OF IMAGE FORMATION
41
where ρ is the electron cloud density and φnuc is the potential of the nucleus given by:
 1 Ze   r  2 

 , r  Rnuc
3  

 nuc (r )   2 Rnuc   Rnuc  

Ze
, r  Rnuc

r
(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):
d el
2
2
 f ( )  g ( )
d
(5.12a)
where f(θ) is the direct scattering amplitude given by:
f ( ) 
1
2ik
 (l  1)(e

2 i l 
l 0

 1)  l (e 2i l   1) Pl (cos  )
(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:
g ( ) 
1
2ik
 e

l 0
2 i l 

 e 2i l  Pl1 (cos  )
(5.12c)
with Pl1(cosθ) being the associated Legendre functions. In the above equations k is the
wave number of the projectile given by:
k
1
E ( E  2mc 2 )
c
(5.13)
For calculation simplicity, the Rutherford elastic scattering differential cross section is
usually used and it is given by (Shimizu and Ze-Jun 1992):
d el ; Ruth
d

Z 2e4
4 E 2 (1  cos   2 ) 2
(5.14)
CHAPTER 5: PHYSICS OF IMAGE FORMATION
42
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 p  of the incident particle as (Fano 1963):
d n 
2
2
dp 
p , n V p,0  ( E   E n  E ) 3
u
h
(5.15)
where p, E, and u are the initial momentum, kinetic energy and velocity of the incident
particle in respect, p  , E  and u the corresponding values after the collision and En the
CHAPTER 5: PHYSICS OF IMAGE FORMATION
43
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:
d n 
1
p , n V p,0
4  4 uu 
2
2
p  2 d (cos  )d
(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:
S
E   
0

with
dE
ds
ds
(5.17a)
dE
 1  1.166 E 
 2 e 4 N A
ln 

ds
AE  J 
(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:
J
 9.76  58.8Z 1.19
Z
(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.
(E’, p’)
θ
(E, p)
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’).
CHAPTER 5: PHYSICS OF IMAGE FORMATION
44
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:
Q
q2
(relativistic)
Q(1 
)
2m
2mc 2
(5.19a)
q2
Q=
(non-relativistic)
2m
(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:
q 2  p 2  p 2  2 pp cos
(5.20)
5.4.2.3. Bethe’s theory revisited
Inokuti (1971) has rewritten equation (5.16) for the non-relativistic case as:
d n  2 z 0 e 4 (mu 2 ) 1 Q 1  n (K ) d (ln Q)
2
2
(5.21)
In this equation:
o K is related to the momentum transfer q=  K=  (k-k’)=p-p’.
o
(K ) 2
Q
is the recoil energy.
2m
o The factor 2 z0 e 4 (mu 2 ) 1 Q 1d (ln Q) is the Rutherford cross section for the
2
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.
CHAPTER 5: PHYSICS OF IMAGE FORMATION
45
o εn(K) is a matrix element related to q. The factor  n (K ) accounts for the fact that if
2
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 19th 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:
z
Fo
e
e it
2
2
m  s    i s 
(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:
ez it e 2
1
a( ) 
e

2
Fo
m  s   2  i s 
(5.23)
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
CHAPTER 5: PHYSICS OF IMAGE FORMATION
46
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):
En
2
 n (K )
Q
(5.24a)
En
2
(Kao ) 2  n (K )
R
(5.24b)
GOS  f n (K ) 
or
f n (K ) 
with
2
me 4
ao 
( Bohr radius ), R 
( Rydberg Energy) . The optical oscillator
me 2
2 2
strength (OOS) is defined as:
lim f n (K )  OOS
K 0
(5.25)
When one deals with excitation to continua (i.e. with ionization) the excitation energy E n
~
is not a discrete variable, but it is a continuous variable E that takes all real values greater
than the first ionization threshold. Then the inelastic cross section for excitation to
d ~
d
~
~
~
continuum states between E and E +d E is ~ dE . Thus ~ is the differential inelastic
dE
dE
~
cross section per E . In this case the GOS is defined as (Inokuti 1971):
~
~
2
df (K, E ) E
(5.26)
GOS 
 (Kao ) 2   E~ , (K )
~
R
dE

~
The final continuum state is specified by E 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
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.
CHAPTER 5: PHYSICS OF IMAGE FORMATION
47
In (Q, W) representation the GOS and OOS are defines as:
GOS 
df (Q,W )
df (0, W )
and OOS 
dW
dW
(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):
E
n
n
 n (K) / Q   f n (K)  Z
2
(5.28)
n
whereas for the continuum case and in the (Q, W) representation (Salvat et al 2003):

df (Q,W )
dW  Z for any Q
dW
0

(5.29)
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):
CHAPTER 5: PHYSICS OF IMAGE FORMATION
48


d 2 inel 2e 4 df (Q,W ) 
 2 sin 2  rW 2mc 2 
2mc 2




dWdQ mu 2
dW WQ (Q  2mc 2 ) (Q(Q  2mc 2 )  W 2 ) 2 
 




longitudinal
transverse
(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:
W 2 /  2  Q(Q  2mc 2 )  W 2
1 
cos  r 
Q(Q  2mc 2 ) 
2W ( E  mc 2 )
2



2
(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  (k ,  ) , 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 q  k and the energy transfer as W=  and
therefore  (Q,W ) . 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:
 1 
df (Q,W )
Q  mc 2 2Z
W
Im

2
2
dW
mc
  p  (Q,W ) 
(5.32)
where Ωp is the plasma energy of a free-electron gas given as:
 p  4 NZ 2 e 2 / m
(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):
CHAPTER 5: PHYSICS OF IMAGE FORMATION
49

  2 sin 2  rW 2mc 2

d 2 inel 2 e 4 df (Q, W ) 
2mc 2



 D(Q,W )  (5.34)


2
2
2
2 2
dWdQ
dW 
mu

WQ (Q  2mc )  (Q(Q  2mc )  W )

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.
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:
CHAPTER 5: PHYSICS OF IMAGE FORMATION
D 
50
 BT
4ne 2
(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:
2u o2 
ne 4 
 dE 

ln 1 

 
2

Eo
 ds  coll
  v  Av 
(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:

E o 

2u o2 

ne ln 1 
2

  v  Av 
(5.37)
4
where    p and ωp is the plasma frequency. The induction of collective oscillations
from a passing electron does no produce angular deflection to the electron.
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 W
required to create an electron-hole pair has been shown to depend on the energy bandgap
Eg via the Klein rule (Klein 1968):
CHAPTER 5: PHYSICS OF IMAGE FORMATION
51
W  2.8E g  E phonon
(5.38)
The phonon term Ephonon accounts for energy losses (like phonon production) and is
typically very small (~0.5 eV). Thus, practically W  2.8 E g . Que and Rowlands (1995b)
have shown that for the case of amorphous semiconductors Klein’s rule is written as:
W  2.2 E g  E phonon
(5.39)
For the case of a-Se (Eg~2.2. eV) one would expect W ~5 eV. Nevertheless, experimental
measurements have found that W >>5 eV. In addition, experiments show that W
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:
CL 
e h
 0 r
(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.
CHAPTER 5: PHYSICS OF IMAGE FORMATION
52
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:
A m D m n
n 0 m 0 m! (m  n)!


p(r , , F )  e  A D 
where A 
(5.41)
e2
eFr (1  cos  )
and ε is the relative dielectric constant, kB the
, D
 rk BT
2k B T
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):
g ( r , ) 
1
4 ro
2
 (r  ro )
(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:
l
n   p(r , , F ) g (r , )d r  n  e
3
where
a
eFro
e2
, b
 ro k BT
k BT
 a b
1   b 2
 l  I l (2 ab )
b l 1  a 
(5.43)
and Il are the modified Bessel functions. The
photogeneration efficiency increases as ε, ro and T increase, whereas n and W 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:
CHAPTER 5: PHYSICS OF IMAGE FORMATION
2
r
t o 
D
53
h  E g 
e2
 eFro cos 
 ro
h p2
(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:
g (r ) 
where G ( E k )  const  e
E
 k
 Ec




G( Ek ) dEk
4 r 2 dr
(5.45)
2
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:
k T
n  4 B
eF
where A 
i

 F 2
i 1  A B
i
dr
 I i (2 AB )   g (r )r e

e

i 1 
0

(5.46)
e2
eFr
. To a first order with respect to F (5.46) is written as:
, B
 rk BT
k BT


e3 F 
n  4 1 

g (r ) r 2 e  A dr
2 2 

2

k
T
B

 0
(5.47)
The thermalization distance r is obtained from:
r2

D
where D 
 k BT
e
Ek 
e2
 eFr
r
t
h p2
(5.48)
and for a-Se: μ = μh = 0.14 cm2/Vs and hνp2=15 meV. The n and W 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.
CHAPTER 5: PHYSICS OF IMAGE FORMATION
54
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.
Formed charge
column
Penetrating electron’s
track
x
z
y
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:
r2
N  b2
n(0) 
e
 b2
(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
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:
0
N esc
 
No
o
e  d
aN o  o
1
ln
8 D 
(5.50)
CHAPTER 5: PHYSICS OF IMAGE FORMATION
with  
55
R2
R2
4 DR 2
,


,
d



dt and No the initial number of
o
4 Dt  b 2
b2
(4 Dt  b 2 ) 2
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:
Y// ( F ) 
 Fb 2
2 Dd
c1e
where μ is the charge mobility and c1 
 c1
 c1 
  c2 e 

e


d


d


 





8 D
8 D
and c2 
 ln
aN o
aN o
2
(5.51)
d2
 b2
F
b2
5.5.2.3. Field perpendicular to the column
In this case the fraction of electrons that escape recombination is:
Y ( F ) 
1
aN o 
1
S ( )
8 D z
(5.52)
with
S ( ) 
1



0
e  s ds

s
s1 
 



, 
2 2 F 2 (h  b 2 ) 2
b2 2 F 2
,
, h  4 Dt  b 2
s

2
2
h
16 D
2D
5.5.2.4. Field at an angle φ with the track
The fraction of electrons that escape recombination is:
Y ( F ) 
with T 
2 F cos 
d
T

dt
  2 2 F 2 (sin  ) 2 t 2 
0


exp
4 Dt  b 2
aN o t

dt
1
2

2 0
4 Dt  b
(5.53)
d
. It is seen that for small F, Y ( F )  Y// ( F cos  ) , whereas for
2  F cos 
intermediate and large F, Y ( F )  Y ( F sin  ) .
CHAPTER 6: MONTE CARLO SIMULATION
CHAPTER 6
MONTE CARLO SIMULATION
56
CHAPTER 6: MONTE CARLO SIMULATION
57
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.
Random
Numbers
Monte
Carlo
Results
Probability
Distribution
Figure 6.1. General block diagram of a Monte Carlo procedure.
CHAPTER 6: MONTE CARLO SIMULATION
58
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
CHAPTER 6: MONTE CARLO SIMULATION
59
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
CHAPTER 6: MONTE CARLO SIMULATION
60
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

 f ( x)dx  1
ii.

b
iii. P(a<X<b)=  f ( x)dx
a
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:
x
F(x) = P(X  x) =
 f (t )dt
(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.
CHAPTER 6: MONTE CARLO SIMULATION
61
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
x*
*
*
r =F(x )=
 f ( x)dx
(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:
b
i. Check if f(x) is normalized. That is check if
 f ( x)dx  1 .
a
x
ii. Calculate the CDF: F(x) =  f (t )dt .
a
iii. Generate random numbers R.
iv. Let F(x) =R and solve for x as x=F-1(R).
CDF
1
r*
x*
x
x*
x
PDF
f(x*)
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.
CHAPTER 6: MONTE CARLO SIMULATION
62
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.
Uniformly distributed
ordered pairs
f(x)/fo
(x*,R2)
1
Region of
rejection
0
a
Region of
acceptance
b
x
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.
CHAPTER 7: PRIMARY ELECTRON GENERATION MODEL
CHAPTER 7
PRIMARY ELECTRON
GENERATION MODEL
63
CHAPTER 7: PRIMARY ELECTRON GENERATION MODEL
64
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:
E p2
(1  cos  p )
2
mc
Ee 
Ep
1
(1  cos  p )
mc 2
(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 φe
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:
(7.2) e)=
PDF(φ
1
2
(7.2)
CHAPTER 7: PRIMARY ELECTRON GENERATION MODEL
65
Primary
Photon
Photoelectric
Absorption
Incoherent
Scattering
Coherent
Scattering
Scattered
photon
Emitted
photoelectron
Ejected
electron
Scattered
photon
Atomic
Deexcitation
Fluorescent
photon
Auger
electron
Coster-Kronig
electron
Electron Interactions
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:

 e  cot 1 1 

Ep   p
 tan 
mc 2   2



(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,  phx y (E ) , is defined as:
AB
A
B
 ph ( E )  wA ph
( E )  wB ph
(E)
Ax B y
(7.4)
A
B
where  ph
( E),  ph
( E) 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:
CHAPTER 7: PRIMARY ELECTRON GENERATION MODEL
P 
A
ph
A
w A ph
(E)
 ph ( E )
Ax B y
66
(7.5)
then PphA  PphB  1 . Thus, the atom which photoelectrically absorbs the photon is
determined by a Monte Carlo decision, based on the probabilities PphA and PphB .
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 LIII
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
particular, if the total photoelectric cross section at energy E is  ph (E ) and the subshell
s
‘s’ photoelectric cross section is  ph
(E ) then:
CHAPTER 7: PRIMARY ELECTRON GENERATION MODEL
67
s
 ph ( E )   ph
(E)
(7.6)
s
If the subshell photoelectric probability is defined as:
s
 ph
(E)
P 
 ph ( E )
s
ph
then
P
s
ph
(7.7)
 1 . Thus, the subshell ‘s’ that photoelectrically absorbs the photon is
s
determined by a Monte Carlo decision, based on the probabilities Pphs . 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):
dI
sin 2 
 A  cos 2 
4
d
h




cos

1 

2mc 2


(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:
h2 ( ) 
h( )
hmax ( )
(7.10)
where h(θ) is given as:
h( ) 
sin 2 
h


  cos  
1 
2
 2mc

4
(7.11)
CHAPTER 7: PRIMARY ELECTRON GENERATION MODEL
68
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).
CHAPTER 7: PRIMARY ELECTRON GENERATION MODEL
69
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).
Element
Zn
Ga
Ge
As
Se
Br
Cd
Te
I
FK
0.870
0.869
0.867
0.866
0.864
0.864
0.841
0.836
0.835
ωK
0.466
0.497
0.528
0.557
0.596
0.613
0.843
0.879
0.886
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 CosterKronig yields (ckLI, ckLII) of subshells LI, LII and LIII for Br, Cd, Te, I, Hg, Tl and Pb.
Element
Br
Cd
Te
I
Hg
Tl
Pb
Fluorescence yield
ωLI
ωLII
ωLIII
0.003
0.019
0.019
0.021
0.059
0.061
0.040
0.079
0.081
0.043
0.085
0.086
0.086
0.376
0.330
0.091
0.390
0.341
0.098
0.404
0.352
aLI
0.185
0.298
0.438
0.432
0.146
0.147
0.148
Auger yield
aLII
aLIII
0.896
0.981
0.778
0.939
0.747
0.919
0.740
0.914
0.497
0.670
0.486
0.659
0.475
0.648
Coster-Kronig yield
ckLI
ckLII
0.812
0.085
0.681
0.163
0.522
0.174
0.525
0.175
0.767
0.126
0.762
0.124
0.753
0.122
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
CHAPTER 7: PRIMARY ELECTRON GENERATION MODEL
70
occurs is determined by a Monte Carlo decision based on the probabilities Psj 
where y  {ω, a, ck} and y s   y sj , requiring
j
P
sj
y sj
ys
,
 1.
j
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 for
K and L shells and the simulated atomic transitions with the corresponding energies and
probabilities for the case of I.
Probability of
Atomic Deexcitation
FKωΚ
0.740
1- FKωΚ
0.260
ωLI
0.043
αLI
0.432
ckLI
ωLII
0.525
0.085
αLII
0.740
ckLII
ωLIII
0.175
0.086
αLIII
0.914
Simulated Atomic
Transition
KLIII
KMIII
KNIII
KLILI
KLIILIII
KLIIILIII
KLIMI
KLIIIMIII
KLIIINIII
KMIIMIII
KMIIINIII
LILIII
LIMIII
LINIII
LIMIVMV
LIMIVNV
LINIVNV
LILIIINV
LIIMI
LIIMIV
LIIMIMII
LIIMIVMV
LIIMIVNV
LIILIIINIV
LIIIMV
LIIINV
LIIIMIVMV
LIIINIVNV
LIIIMIIIMIII
Transition's
Probability
0.820
0.147
0.033
0.078
0.466
0.124
0.028
0.234
0.037
0.024
0.008
0.009
0.806
0.186
0.797
0.198
0.005
1.000
0.047
0.953
0.038
0.819
0.142
1.000
0.880
0.120
0.781
0.016
0.203
Transition's
Energy (keV)
28.612
32.294
33.046
22.793
23.760
24.055
26.909
27.737
28.489
31.363
32.171
0.631
4.313
5.065
3.938
4.508
5.089
0.582
3.780
4.221
2.849
3.602
4.172
0.244
3.938
4.508
3.307
4.458
4.557
CHAPTER 7: PRIMARY ELECTRON GENERATION MODEL
71
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:
P( ,  )dd 
1
1
sin d
d
2
2
(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
CHAPTER 7: PRIMARY ELECTRON GENERATION MODEL
72
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.
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
73
CHAPTER 8
PRIMARY ELECTRON GENERATION:
RESULTS & DISCUSSION
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
74
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.
Spectral Group
(Anode Material/
1 Filter/ 2nd Filter)
st
Mo/ 0/ 0
Spectra
(Anode Material/
1 Filter/ 2nd Filter)
st
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
40 kVp Mo/ 0/ 0
30 kVp W/ 1.02mm Al/ 0
30kVp Mo/ 0.51mm Al/ 0
Mo/ Al/ Mo
W/ Al/ 0
30 kVp W/ 3.05mm 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
40kVp Mo/ 1.52mm Al/ 0
40 kVp W/ 2.03mm Al/ 0
30kVp Mo/ 0.51mm Al/ 0.030mm Mo
40 kVp W/ 3.05mm Al/ 0
35kVp Mo/ 0.51mm Al/ 0.030mm Mo
40kVp Mo/ 0.51mm Al/ 0.030mm Mo
30 kVp W/ 0/ BS 0.13mm of SN
W/ 0/ BS*
25kVp MoW/ 0.51mm Al/ 0
25kVp Mo/ 0/ 0.030mm Mo
25kVp Mo/ 0/ 0.015mm Mo
30kVp MoW/ 0.51mm Al/ 0
30kVp Mo/ 0/ 0.015mm Mo
30kVp Mo/ 0/ 0.030mm Mo
40 kVp W/ 0/ BS 0.13mm of SN
40 kVp W/ 0/ BS 0.13mm of LA
20kVp Mo/ 0/ 0.030mm Mo
W/ 0/ 0
30 kVp W/ 2.03mm Al/ 0
30kVp Mo/ 1.02mm Al/ 0
40kVp Mo/ 1.02mm Al/ 0.030mm Mo
Mo/ 0/ Mo
Spectra
(Anode Material/
1 Filter/ 2nd Filter)
st
20 kVp Mo/ 0/ 0
25kVp Mo/ 0.51mm Al/ 0
Mo/ Al/ 0
Spectral Group
(Anode Material/
1 Filter/ 2nd Filter)
st
30kVp MoW/ 1.02mm Al/ 0
MoW/ Al/ 0
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
40kVp Mo/ 0/ 0.030mm Mo
40kVp MoW/ 2.03mm Al/ 0
25 kVp W/ 0/ 0
40kVp MoW/ 1.52mm Al/ 0
30 kVp W/ 0/ 0
35 kVp W/ 0/ 0
40 kVp W/ 0/ 0
MoW/ Al/ Mo
MoW/ 0/ Mo
30kVp MoW/ 0.51mm Al/ 0.030mm Mo
40kVp MoW/ 0.51mm Al/ 0.030mm Mo
30kVp MoW/ 0/ 0.030 mm Mo
*
Beam Shaping
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
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.
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
75
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
photoelectrons, Auger and CK electrons. The energy distributions of primary electrons
are characterised from the presence of
certain spectral peaks which are due to
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
3
3.5
3
2.5
2
1.5
1
0.5
0
2.5
2
1.5
1
0.5
Number of fluorescent photons x 10
3
Number of fluorescent photons x 10
Number of fluorescent photons x 10
4
6
6
3.5
6
4.5
76
0
2.5
2
1.5
1
0.5
0
19 21 23 25 27 29 31 33 35 37 39 41
19 21 23 25 27 29 31 33 35 37 39 41
19 21 23 25 27 29 31 33 35 37 39 41
(a) 1 3 5 7 9 11 13 15 17Energy
(b) 1 3 5 7 9 11 13 15 17Energy
(c) 1 3 5 7 9 11 13 15 17Energy
(keV)
(keV)
(keV)
6
4
4
2
1.5
1
0.5
0
3.5
3
2.5
2
1.5
1
0.5
3.5
Number of fluorescent photons x 10
Number of fluorescent photons x 10
Number of fluorescent photons x 10
6
4.5
6
3
2.5
0
3
2.5
2
1.5
1
0.5
0
19 21 23 25 27 29 31 33 35 37 39 41
19 21 23 25 27 29 31 33 35 37 39 41
19 21 23 25 27 29 31 33 35 37 39 41
(d) 1 3 5 7 9 11 13 15 17Energy
(e) 1 3 5 7 9 11 13 15 17Energy
(f) 1 3 5 7 9 11 13 15 17Energy
(keV)
(keV)
(keV)
2
1.5
1
0.5
0
1.5
6
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
Number of fluorescent photons x 10
6
Number of fluorescent photons x 10
Number of fluorescent photons x 10
6
3
2.5
1
0.5
0
19 21 23 25 27 29 31 33 35 37 39 41
19 21 23 25 27 29 31 33 35 37 39 41
19 21 23 25 27 29 31 33 35 37 39 41
(g) 1 3 5 7 9 11 13 15 17Energy
(h) 1 3 5 7 9 11 13 15 17Energy
(i) 1 3 5 7 9 11 13 15 17Energy
(keV)
(keV)
(keV)
4.5
6
6
4.5
6
1.5
1
0.5
3
2.5
2
1.5
1
0.5
0
0
(j)
3.5
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41
Energy (keV)
4
Number of fluorescent photons x 10
Number of fluorescent photons x 10
Number of fluorescent photons x 10
4
3.5
3
2.5
2
1.5
1
0.5
0
19 21 23 25 27 29 31 33 35 37 39 41
19 21 23 25 27 29 31 33 35 37 39 41
(k) 1 3 5 7 9 11 13 15 17Energy
(l) 1 3 5 7 9 11 13 15 17Energy
(keV)
(keV)
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
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).
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
Number of primary photons x 10
6
Number of primary photons escaping x 10
3
2.5
2
1.5
1
0.5
7
6
Forwards
Backwards
Cd K-edge
5
4
3
Te K-edge
2
1
0
0
(a)
77
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 3739 41
Energy (keV)
(b)
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41
Energy (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
Number of fluorescent photons escaping x 10
3
escape forwards and backwards in CdTe.
200
180
160
140
120
Forwards
Backwards
100
80
60
40
20
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41
Energy (keV)
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
been chosen in order to let all the possible atomic transitions to occur and thus the
corresponding spectral peaks to be presented.
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
6
5
4
3
2
1
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41
Energy (keV)
6
5
4
3
2
1
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41
Energy (keV)
6
3
2
1
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41
Energy (keV)
(c)
10
Number of primary electrons x 10
8
7
6
5
4
3
2
1
0
(e)
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41
Energy (keV)
5
4
3
2
1
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41
Energy (keV)
1 3 5 7 9 11 1315 17 19 21 2325 27 29 31 3335 37 39 41
Energy (keV)
12
11
10
9
8
7
6
5
4
3
2
1
0
(f)
6
10
9
8
7
6
(h)
6
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41
Energy (keV)
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
(i)
8
1 3 5 7 9 11 13 1517 19 2123 25 27 2931 33 3537 39 41
Energy (keV)
8
6
6
7
3
Number of primary electrons x 10
Number of primary electrons x 10
4
Number of primary electrons x 10
Number of primary electrons x 10
10
9
8
7
6
3.5
2.5
2
1.5
1
0.5
0
(j)
5
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
1 3 5 7 9 1113 15 17 1921 23 2527 29 3133 35 37 3941
Energy (keV)
7
Number of primary electrons x 10
6
Energy (keV)
(g)
6
9
1 3 5 7 9 11 1315 17 1921 23 2527 29 3133 35 3739 41
(d)
7
(b)
6
6
Number of primary electrons x 10
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
8
Number of primary electrons x 10
7
(a)
Number of primary electrons x 10
Number of primary electrons x 10
8
Number of primary electrons x 10
Number of primary electrons x 10
9
6
6
6
10
78
6
5
4
3
2
1
0
(k)
6
5
4
3
2
1
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41
Energy (keV)
(l)
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41
Energy (keV)
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
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
of photons have energies lower than the binding energies of Cd and Te K shells
(26.711 keV and 31.814 keV respectively).
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
5.5
8
Primary photons
Primary electrons
5
Primary photons
Primary electrons
6
7
Number of paricles x 10
6
4.5
Number of paricles x 10
79
4
3.5
3
2.5
2
1.5
6
5
4
3
2
1
1
0.5
0
(a)
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41
Energy (keV)
(b)
4
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41
Energy (keV)
4
Primary photons
Primary electrons
Primary photons
Primary electrons
6
3.5
Number of paricles x 10
Number of paricles x 10
6
3.5
3
2.5
2
1.5
1
0.5
2.5
2
1.5
1
0.5
0
(c)
3
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41
Energy (keV)
(d)
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41
Energy (keV)
2.25
Primary photons
Primary electrons
Number of paricles x 10
6
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0
(e)
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41
Energy (keV)
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
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
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
80
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, kVp: 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
process, PDF(φ) = cos 2  (Sakellaris et al 2005), an expected fact since in the
mammographic energy range the photoelectric effect dominates. Thus, the primary
Number of primary electrons x 103
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
85
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0
81
32 keV
27 keV
10 keV
5 keV
4 keV
0
0.5
1
1.5
2
2.5
3
3.5
4
Azimuthal angle φ (rad)
4.5
5
5.5
6
2 keV
3 keV
4 keV
5 keV
6 keV
7 keV
8 keV
9 keV
10 keV
11 keV
12 keV
13 keV
14 keV
15 keV
16 keV
17 keV
18 keV
19 keV
20 keV
21 keV
22 keV
23 keV
24 keV
25 keV
26 keV
27 keV
28 keV
29 keV
30 keV
31 keV
32 keV
33 keV
34 keV
35 keV
36 keV
37 keV
38 keV
39 keV
40 keV
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
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
1
0.5
0.9
0.45
Minima of normalized azimuthal
distributions
Normalized number of primary electrons
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
0.8
0.7
0.6
CdTe
0.5
0.4
HgI2
0.3
a-Se
82
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0.2
(a)
0
0.5
1
1.5
2
2.5
3
3.5
4
Azimuthal angle φ (rad)
4.5
5
5.5
6
0
(b)
a-Se
a-As2Se3
Ge
PbO
GaSe
GaAs
PbI2
HgI2
TlBr
ZnTe
CdZnTeCd08Zn02Te CdTe
Material
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
polyenergetic spectra, a-Se, a-As2Se3 and Ge have the minimum azimuthal uniformity
whereas CdZnTe, Cd0.8Zn0.2Te and CdTe the maximum one.
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
Number of primary electrons x 10 3
Number of primary electrons x 103
7
6
5
4
3
2
Most probable cosθ
1
-1
-0.8 -0.6 -0.4 -0.2
0
0.2
cosθ
0.4
0.6
0.8
1
5
4
3
2
Most probable cosθ
1
-1
-0.8 -0.6 -0.4 -0.2
0 0.2
cosθ
0.4
0.6
0.8
1
0.6
0.8
1
8
Number of primary electrons x 103
Number of primary electrons x 103
6
(b)
8
7
6
5
4
3
2
Most probable cosθ
1
0
(c)
7
0
0
(a)
83
7
6
5
4
3
2
Most probable cosθ
1
0
-1
-0.8 -0.6 -0.4 -0.2
Figure 8.8. The
0 0.2
cosθ
0.4
0.6
0.8
1
(d)
-1
polar distributions (cosθ) of
-0.8 -0.6 -0.4 -0.2
0
0.2
cosθ
0.4
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.
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
Number of primary electrons x 103
Number of primary electrons x 103
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
12
10
8
6
4
Most probable cosθ
2
0
-1
-0.8 -0.6 -0.4 -0.2
0 0.2
cosθ
0.4
0.6
0.8
1
14
12
10
8
6
4
Most probable cosθ
2
0
(c)
14
12
10
8
6
4
Most probable cosθ
2
0
-1
-0.8 -0.6 -0.4 -0.2
0 0.2
cosθ
0.4
(b)
Number of primary electrons x 10 3
Number of primary electrons x 103
(a)
84
0.6
0.8
1
-1
-0.8 -0.6 -0.4 -0.2
0 0.2
cosθ
0.4
0.6
0.8
1
0.8
1
14
12
10
8
6
4
Most probable cosθ
2
0
(d)
-1
-0.8 -0.6 -0.4 -0.2
0 0.2
cosθ
0.4
0.6
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
CK electrons are isotropically ejected (the normalized polar PDF=sinθ= 1 cos 2  ), 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.
Number of primary electrons x 103
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
85
12
10
8
6
4
2
0
-1
-0.8 -0.6 -0.4 -0.2
0 0.2
cosθ
0.4
0.6
0.8
1
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
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.
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
86
-150
-75
y (µm) 0
75
150
(a)
-150
-75
0
x (µm)
75
150
(b)
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:
  x  b 2 
1
 
f ( x)  a1 exp   
  c1  


(8.1)
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= 2 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
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
87
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
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
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
1.55
1.5
1.5
1.45
1.45
1.4
1.4
1.35
1.35
FWHM (μm)
FWHM (μm)
1.55
1.3
1.25
1.2
1.3
1.25
1.2
1.15
1.15
1.1
1.1
1.05
1.05
1
1
2 4
(a)
6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Energy (keV)
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Energy (keV)
2 4 6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Energy (keV)
1.55
1.5
1.5
1.45
1.45
1.4
1.4
1.35
1.35
FWHM (μm)
FWHM (μm)
4 6
2
(b)
1.55
1.3
1.25
1.2
1.15
1.3
1.25
1.2
1.15
1.1
1.1
1.05
1.05
1
(c)
88
1
2 4
6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Energy (keV)
(d)
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
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
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
89
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
Material
FWHM (μm)
a-Se
1.346
a-As2Se3
1.372
Ge
1.404
CdTe
1.411
TlBr
1.455
Cd0.8Zn0.2Te
1.470
PbI2
1.472
PbO
1.476
HgI2
1.477
GaSe
1.485
CdZnTe
1.492
GaAs
1.507
ZnTe
1.568
Zone B
Material
FWHM (μm)
a-Se
1.345
a-As2Se3
1.372
Ge
1.404
CdTe
1.424
Cd0.8Zn0.2Te
1.437
CdZnTe
1.444
TlBr
1.455
PbI2
1.471
HgI2
1.476
PbO
1.476
GaSe
1.486
GaAs
1.509
ZnTe
1.567
Zone C
Material
FWHM (μm)
a-Se
1.345
a-As2Se3
1.372
Ge
1.403
TlBr
1.453
PbO
1.474
PbI2
1.475
HgI2
1.477
GaSe
1.486
CdTe
1.497
CdZnTe
1.506
ZnTe
1.507
GaAs
1.509
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
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
1.00E+08
1.00E+07
1.00E+07
1.00E+06
1.00E+05
1.00E+04
1.00E+03
1.00E+02
1.00E+04
1.00E+03
1.00E+02
1.00E+00
1.00E+00
Horizontal Position (μm)
(d)
-700 -600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600 700
Horizontal Position (μm)
1.00E+08
1.00E+08
1.00E+07
1.00E+07
1.00E+06
1.00E+05
1.00E+04
1.00E+03
1.00E+02
1.00E+05
1.00E+04
1.00E+03
1.00E+02
1.00E+00
1.00E+00
(b)
1.00E+06
1.00E+01
1.00E+01
-700 -600 -500 -400 -300 -200 -100 0
100 200 300 400 500 600 700
Horizontal Position (μm)
(e)
-700 -600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600 700
Horizontal Position (μm)
1.00E+08
1.00E+08
1.00E+07
1.00E+07
Number of primary electrons
Number of primary electrons
1.00E+05
1.00E+01
-700 -600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600 700
1.00E+06
1.00E+05
1.00E+04
1.00E+03
1.00E+02
1.00E+06
1.00E+05
1.00E+04
1.00E+03
1.00E+02
1.00E+01
1.00E+01
1.00E+00
1.00E+00
(c)
90
1.00E+06
1.00E+01
(a)
Number of primary electrons
Number of primary electrons
1.00E+08
Number of primary electrons
Number of primary electrons
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
-700 -600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600 700
Horizontal Position (μm)
(f)
-700 -600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600 700
Horizontal Position (μm)
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.
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
1.00E+08
1.00E+08
1.00E+07
1.00E+07
1.00E+06
1.00E+05
1.00E+04
1.00E+03
1.00E+02
Number of primary electrons
Number of primary electrons
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
1.00E+01
-600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600
Horizontal Position (μm)
1.00E+04
1.00E+03
1.00E+02
1.00E+00
-600 -500 -400 -300 -200 -100
1.00E+08
1.00E+07
1.00E+07
1.00E+06
1.00E+05
1.00E+04
1.00E+03
1.00E+02
-600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600
Horizontal Position (μm)
1.00E+06
1.00E+05
1.00E+04
1.00E+03
1.00E+02
1.00E+00
(e)
-600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600
Horizontal Position (μm)
1.00E+08
1.00E+07
1.00E+07
1.00E+05
1.00E+04
1.00E+03
1.00E+02
1.00E+01
1.00E+00
(c)
-600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600
Horizontal Position (μm)
Number of primary electrons
1.00E+08
1.00E+06
100 200 300 400 500 600
1.00E+01
1.00E+00
(b)
0
Horizontal Position (μm)
1.00E+08
Number of primary electrons
Number of primary electrons
1.00E+05
(d)
1.00E+01
Number of primary electrons
1.00E+06
1.00E+01
1.00E+00
(a)
91
1.00E+06
1.00E+05
1.00E+04
1.00E+03
1.00E+02
1.00E+01
1.00E+00
(f)
-600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600
Horizontal Position (μm)
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.
(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
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
z (mm) 0
z (mm) 0
0.5
0.5
1
1
(a)
-1
-0.5
0
0.5
1
y (mm)
(d)
z (mm) 0
z (mm) 0
0.5
0.5
1
1
(b)
-1
-0.5
0
0.5
1
y (mm)
(e)
z (mm) 0
z (mm) 0
0.5
0.5
1
1
(c)
-1
-0.5
0
0.5
1
y (mm)
(f)
92
-1
-0.5
0
0.5
1
y (mm)
-1
-0.5
0
0.5
1
y (mm)
-1
-0.5
0
0.5
1
y (mm)
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
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
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
93
60
Grey Color
50
40
30
D50%
20
Dmax
10
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Depth (mm)
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
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.
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
94
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.
Material
R (μm) Dmax (μm) D50% (μm)
PbO
200
320
70
TlBr
300
400
83
Ge
300
500
107
GaSe
300
560
105
GaAs
310
560
98
a-Se
350
540
130
a-As2Se3
350
520
126
HgI2
370
500
157
ZnTe
400
490
157
PbI2
400
490
157
CdZnTe
450
570
190
Cd0.8Zn0.2Te
500
630
192
CdTe
500
660
193
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
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,
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
95
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.
Material
ZnTe
PbI2
HgI2
CdTe
CdZnTe
Cd0.8Zn0.2Te
R (μm) Dmax (μm) D50% (μm)
75
110
33
100
120
51
100
120
46
150
200
51
150
180
50
150
200
51
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.
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.
Category
A
B
C
D
Materials
a-Se, a-As2Se3, GaSe, GaAs, Ge
CdTe, CdZnTe, Cd0.8Zn0.2Te, ZnTe
PbO, TlBr
PbI2, HgI2
600
550
500
450
400
350
300
250
200
150
100
50
0
(a)
Total
Forwards
Backwards
.
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Energy (keV)
Number of primary photons escaping
x 103
Number of primary photons escaping
x 103
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
60
55
50
45
40
35
30
25
20
15
10
5
0
(b)
96
Total
Forwards
Backwards
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Energy (keV)
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.
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 PbI2
(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
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
8
Number of fluorescent photons produced
x 106
Number of fluorescent photons produced
x 106
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
7
6
5
4
3
2
1
0
0
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Energy (keV)
(a)
97
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Energy (keV)
(b)
Figure 8.18. The energy-related number distributions of fluorescent photons produced
6
11
5.5
10
5
4.5
4
a-Se
a-As2Se3
GaSe
GaAs
Ge
3.5
3
2.5
2
1.5
1
Number of fluorescent photons produced x 10 6
Number of fluorescent photons produced x 10 6
in (a) a-As2Se3 and (b) PbI2.
9
8
7
5
4
3
2
1
0.5
0
0
0
(a)
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
2 4
6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Energy (keV)
8
3
2.5
2
PbO
TlBr
1.5
1
0.5
Number of fluorescent photons produced x 106
Number of fluorescent photons produced x 106
0
(b)
Energy (keV)
3.5
7
6
5
PbI2
HgI2
4
3
2
1
0
0
(c)
CdTe
CdZnTe
Cd08Zn02Te
ZnTe
6
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Energy (keV)
(d)
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Energy (keV)
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
(a)
650
600
550
500
450
400
350
300
250
200
150
100
50
0
Total
Forwards
Backwards
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Energy (keV)
Number of fluorescent photons escaping
x 103
Number of fluorescent photons escaping
x 103
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
1500
1400
1300
1200
1100
1000
900
800
700
600
500
400
300
200
100
0
(b)
98
Total
Forwards
Backwards
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Energy (keV)
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.
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.
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
Total
Primary
Fluorescent
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Energy (keV)
(a)
1100
1000
900
800
700
600
500
400
300
200
100
0
Number of escaping photons x 103
Number of escaping photons x 103
1200
1100
1000
900
800
700
600
500
400
300
200
100
0
99
Total
Primary
Fluorescent
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Energy (keV)
(b)
Figure 8.21. The energy-related number distributions of escaping primary and fluorescent
1600
2100
1400
1800
1200
1000
a-Se
a-As2Se3
GaSe
GaAs
Ge
800
600
400
Number of escaping photons x 103
Number of escaping photons x 103
photons in (a) GaSe and (b) PbI2.
0
2
4
6
900
600
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
(b)
Energy (keV)
700
Energy (keV)
1200
600
1000
500
400
PbO
TlBr
300
200
Number of escaping photons x 103
Number of escaping photons x 103
CdTe
CdZnTe
Cd08Zn02Te
ZnTe
0
0
800
PbI2
HgI2
600
400
200
100
0
0
0
(c)
1200
300
200
(a)
1500
2
4
6
0
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Energy (keV)
(d)
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Energy (keV)
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.
(a)
100
45
22
20
18
16
14
12
10
8
6
4
2
0
Number of primary electrons produced
x 106
Number of primary electrons produced
x 106
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
40
35
30
25
20
15
10
5
0
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Energy (keV)
(b)
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Energy (keV)
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.
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.
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
24
45
20
18
16
a-Se
a-As2Se3
GaSe
GaAs
Ge
14
12
10
8
6
4
Number of primary electrons produced x 106
Number of primary electrons produced x 106
22
40
35
30
20
15
10
5
0
0
2
4
6
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
(b)
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Energy (keV)
Energy (keV)
36
22
33
20
18
16
14
PbO
TlBr
12
10
8
6
4
Number of primary electrons produced x 106
Number of primary electrons produced x 106
0
24
30
27
24
21
PbI2
HgI2
18
15
12
2
9
6
3
0
(c)
CdTe
CdZnTe
Cd08Zn02Te
ZnTe
25
2
(a)
101
0
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Energy (keV)
(d)
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Energy (keV)
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.
CHAPTER 8: PRIMARY ELECTRON GENERATION-RESULTS & DISCUSSION
102
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.
Escaping
photons
Direction
Forwards
Backwards
Total
Forwards
Backwards
Total
Forwards
Backwards
Total
Primary
Fluorescent
Primary &
Fluorescent
Average
Percentage
(%)
0.2
0.2
0.4
0.02
7.5
7.5
0.2
7.7
7.9
Maximum
Percentage
(%)
5.5
0.6
5.9
0.4
30.7
30.7
5.7
30.8
30.8
Material
GaSe 40 keV
CdTe 26 keV
GaSe 40 keV
a-Se 40 keV
a-Se 13 keV
a-Se 13 keV
GaSe 40 keV
a-Se 13 keV
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).
Energy
(keV)
Number of
fluorescent photons
min
max
Number of
escaping photons
min
max
Number of
primary electrons
min
max
16-26
CdTe
GaSe, GaAs
CdTe,
CdZnTe,
Cd0.8Zn0.2Te,
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
a-Se,
a-As2Se3
PbO, a-Se
TlBr, GaAs,
GaSe, ZnTe
CHAPTER 9: A PRELIMINARY STUDY ON FINAL SIGNAL FORMATION IN a-Se
103
CHAPTER 9
A PRELIMINARY STUDY ON FINAL
SIGNAL FORMATION IN a-Se
CHAPTER 9: A PRELIMINARY STUDY ON FINAL SIGNAL FORMATION IN a-Se
104
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.
ii. A uniform electric field of the form E 
Vtop el
d a  Se
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.
x
Top Electrode
F
Uiy
Uix
y
F
Uf
VACUUM
Ui
E
Uiz
z
z-Global System
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.
CHAPTER 9: A PRELIMINARY STUDY ON FINAL SIGNAL FORMATION IN a-Se
105
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:
m
dU
 F  eE
dt
(9.1)
whereas the theorem for kinetic energy change is:
E k ; f  E k ;i  Wel . f

z f 0
f
1
2
2
m(U f  U i )  e  Edr  eE ( z f  z i )  eEzi
i
2
(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:
Ek ; f  Ek ;i  eVtop el
zi
(9.3)
d a  Se
Position of primary electrons:
xf
2 Ek ;i
yf 
2 E k ;i
m
m
sin  i cos i t f  xi
(9.4a)
sin  i sin i t f  yi
(9.4b)
zf  0
(9.4c)
Drifting time of primary electrons:
t f 1, 2 
2 E k ;i
md a  Se
eVtop el
cos  i 
m
 2md a  Se
8md a  Se

zi  
 eVtop el
eVtop el

1

2

cos  i 

m

2 E k ;i
(9.5a)
2
(9.5b)
Direction of primary electrons:
 f  i
2 E k ;i
cos  f 
m
cos  i 
2 E k ;i
m

(9.6)
eVtop el
md a  Se
2eVtop el
md a  Se
zi
tf
(9.7)
CHAPTER 9: A PRELIMINARY STUDY ON FINAL SIGNAL FORMATION IN a-Se
106
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.
6
Number of primary electrons x 10
8
Initial energies
Energies on top electrode
7
6
5
4
3
2
1
0
1 3 5
7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
Energy(keV)
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.
CHAPTER 9: A PRELIMINARY STUDY ON FINAL SIGNAL FORMATION IN a-Se
107
Number of primary electrons
x 103
200
180
160
140
120
100
80
60
40
20
0
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
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,
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
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).
CHAPTER 9: A PRELIMINARY STUDY ON FINAL SIGNAL FORMATION IN a-Se
108
-3
-2.4
-1.8
-1.2
-0.6
y (mm)
0
0.6
1.2
1.8
2.4
3
-3 -2.4 -1.8 -1.2 -0.6 0 0.6 1.2 1.8 2.4
x (mm)
3
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.
160
Number of primary electrons x 103
Number of primary electrons x 103
160
140
120
100
80
60
40
20
140
120
100
80
60
40
20
0
(a)
0
-3
-2.4 -1.8 -1.2 -0.6
0
0.6
x (mm)
1.2
1.8
2.4
3
(c)
1.00E+05
1.00E+04
1.00E+03
1.00E+02
1.00E+01
1.00E+00
(b)
-2.4 -1.8 -1.2 -0.6
0
0.6
x (mm)
1.2
1.8
2.4
3
1.00E+06
Number of primary electrons
Number of primary electrons
1.00E+06
-3
1.00E+05
1.00E+04
1.00E+03
1.00E+02
1.00E+01
1.00E+00
-3 -2.4 -1.8 -1.2 -0.6 0 0.6 1.2 1.8 2.4
x (mm)
3
(d)
-3 -2.4 -1.8 -1.2 -0.6 0 0.6 1.2 1.8 2.4
y (mm)
3
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.
Number of primary electrons x 103
CHAPTER 9: A PRELIMINARY STUDY ON FINAL SIGNAL FORMATION IN a-Se
109
Initial distribution
Distribution on top electrode
250
200
150
100
50
0
0
0.5
1
1.5
2
2.5
3
Polar angle θ (rad)
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.
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
Polar angle θ (rad)
frequency of appearance of the different (θ, φ) pairs.
Azimuthal angle φ (rad)
Figure 9.6. The grey scale image representing the frequency of appearance of the
different (θ, φ) pairs.
CHAPTER 10: ELECTRIC FIELD CONSIDERATIONS IN a-Se
CHAPTER 10
ELECTRIC FIELD CONSIDERATIONS
IN a-Se
110
CHAPTER 10: ELECTRIC FIELD CONSIDERATIONS IN a-Se
111
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
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 Vtop 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.
CHAPTER 10: ELECTRIC FIELD CONSIDERATIONS IN a-Se
ITO Top electrode
112
Vtop el=5000 V
x
σgap(x,y)
Active Matrix
++++
da-Se=500 μm
a-Se
z
++++
++++
Vp~10 V
++++
++++
++++
dSiO2=1-5 μm
SiO2
Insulator
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 Vtop 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 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  gap ( x, y ) can be considered
in the gaps.
Pang et al solved the Laplace’s equation in three dimensions:
 2V ( x, y, z )  0
(10.1)
in both the a-Se and the insulator layers with the following boundary conditions:
V ( x, y, z) z 0  Vtopel , V ( x, y, z) z d
V ( x, y , z ) z  d

 d a  Se
 V ( x, y , z ) z  d
 0,
a  S e  d a  S e
(10.2)
(10.3)
If (x,y) is on the pixel electrodes
V ( x, y , z ) z  d

a  S e d a  S e
SiO2
a  Se d a  Se
 V ( x, y , z ) z  d
a  Se d a  Se
 Vp
(10.4)
If (x,y) is in the gap region
 a  Se
V ( x, y, z )
z
z  d a  Se d a  Se
  SiO2
V ( x, y, z )
z
  gap ( x, y )
(10.5)
z  d a  Se d a  Se
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,
CHAPTER 10: ELECTRIC FIELD CONSIDERATIONS IN a-Se
113
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:
V ( x, y, z )
z
  E z ( x, y )
(10.6)
z  d a  Se d a  Se
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 E  V .
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
(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).
Ty
Tx
(a)
(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:
V ( x, y, z )  V1  A00 z / D 

A
m , n 0
mn
sinh( 2 (m / Tx ) 2  (n / Ty ) 2 z ) 
 cos( 2  mx / Tx ) cos( 2  ny / Ty ) for 0  z  D
(10.7)
CHAPTER 10: ELECTRIC FIELD CONSIDERATIONS IN a-Se

114
Insulator layer:
V ( x, y, z )  B00 ( z  D  d ) / D 

B
m , n 0
mn
sinh( 2 (m / Tx ) 2  (n / Ty ) 2 ( z  D  d )) 
 cos( 2  mx / Tx ) cos( 2  ny / Ty ) for D  z  D  d
(10.8)
The coefficients A00, Amn, B00 and Bmn are calculated as follows:
From equation (10.3) the relation below is obtained:
when m  n  0
 (V1 D  A00 D) / d ,



sinh( 2 (mD / Tx ) 2  (nD / Ty ) 2 )
 Amn
, otherwise
2
2

sinh(

2

(
md
/
T
)

(
nd
/
T
)
)
x
y

(10.9)
when m  n  0
 A00 ,
Cmn  
2
2
 Amn sinh( 2 (mD / Tx )  (nD / Ty ) ), otherwise
(10.10)
Bmn
It is defined:
which satisfies the following equation:

C
m , n 0
mn
g mn ( x, y) cos( 2  mx / Tx ) cos(2  ny / Ty )  S ( x, y)
(10.11)
The expressions for g mn ( x, y) and S ( x, y ) 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 σgap(x,y) is known:
when ( x, y) is on the pixel electrodes
1,

(10.12)
 D
g mn ( x, y)  
 mn (1   m0 )(1   n 0 )  (1  2 ) m0 n 0 , when ( x, y) is in the gap region

1 d
when ( x, y) is on the pixel electrodes
V0  V1 ,


D
S ( x, y)  
 gap ( x, y) D / 1  2 V1 , when ( x, y) is in the gap region

1 d

 mn  2 
(10.13)
 cosh( 2  (mD / T ) 2  (nD / T ) 2 )
x
y

(mD / Tx )  (nD / T y ) 

2
2
sinh(
2

(
mD
/
T
)

(
nD
/
T
)
)

x
y

2
2
 cosh( 2  (md / Tx )  (nd / T y ) ) 
 2

1 sinh( 2  (md / Tx ) 2  (nd / T y ) 2 ) 

2
2
(10.14)
CHAPTER 10: ELECTRIC FIELD CONSIDERATIONS IN a-Se

115
If Ez(x,y) is known:
when ( x, y ) is on the pixel electrodes
1,
g mn ( x, y )  
 (1   )(1   )    , when ( x, y ) in the gap region
m0
n0
m0 n 0
 mn
(10.15)
when ( x, y ) is on the pixel electrodes
V  V1 ,
S ( x, y )   0
 Ez ( x, y ) D, when ( x, y ) is in the gap region
(10.16)
 cosh( 2  (mD / T ) 2  (nD / T ) 2 ) 
x
y


(mD / Tx )  (nD / T y ) 

2
2
 sinh( 2  (mD / Tx )  (nD / T y ) ) 
 mn  2 
2
2
(10.17)
 2 mx   2 ny 
 and integrating
 cos
Multiplying both sides of equation (10.11) by cos
 T 
T
x

  y 
over x and y, it is obtained:

E
m , n 0
mn
mn
C mn  I mn
m, n  0,1, 2,...,
(10.18)
where
I mn 
Tx / 2
Ty / 2

0
S ( x, y) cos( 2  mx / Tx ) cos( 2  ny / Ty )
(10.19)
0
Emmnn 
Tx / 2
Ty / 2

0
g mn ( x, y) cos( 2  mx / Tx ) cos( 2  ny / Ty ) cos( 2  mx / Tx ) cos( 2  ny / Ty ) (10.20)
0
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

N max
in (10.18) by
m ,n 0

, where Nmax is an integer.
m ,n 0
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):
V ( x, y, z )  V1  A00 z / D 
N max

m , n 0
Amn sinh( 2 (m / Tx ) 2  (n / Ty ) 2 z ) 
 (m)(n) cos( 2  mx / Tx ) cos( 2  ny / Ty ) for 0  z  D
(10.22)
CHAPTER 10: ELECTRIC FIELD CONSIDERATIONS IN a-Se
116
where
when n  0
1,

 
n 


(n)   sin 
N max  1  , otherwise


n

 N 1
max

(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.
4
V(x, y, z) x 10 V
µm
µm
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 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.
CHAPTER 11: MODEL FORMULATION FOR ELECTRON INTERACTIONS IN a-Se
CHAPTER 11
MODEL FORMULATION FOR
ELECTRON INTERACTIONS IN a-Se
117
CHAPTER 11: MODEL FORMULATION FOR ELECTRON INTERACTIONS IN a-Se
118
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:
P( s)   e
1
tot

s
tot
(11.1)
where λtot is the total mean free path. The number of molecules (atoms) per unit volume is:
N  NA

AM
(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:
 tot( E )   el ( E )   inel ( E )
(11.3)
where σel and σinel are the elastic and inelastic scattering cross sections respectively. Thus,
the mean free path is defined as:
tot 
1
N tot
(11.4)
CHAPTER 11: MODEL FORMULATION FOR ELECTRON INTERACTIONS IN a-Se
119
11.3. Decision on the type of electron interaction
From equation (11.3) it is derived that
 el  inel

 1 . If the probabilities for elastic (Pel)
 tot  tot
and inelastic (Pinel) scattering are defined as:
Pel 
 el
 tot
(11.5a)
Pinel 
 inel
 tot
(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):
d el
2
 f ( )
d
(11.6a)
where
f ( ) 
1
2ik

 (2l  1)(e
2 il
l 0
 1) Pl (cos  )
(11.6b)
As it is stated by Salvat et al (1985) the scattering amplitude f ( ) can be calculated by
2
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):

sin( qr )
V (r )r 2 dr
qr
0
f ( B ) ( )  2
(11.7)
where q  2k sin(  / 2) is the momentum transfer. Equation (11.7) can be written as:
f ( B ) ( ) 
1
2ik

 (2l  1)2i
l o
( B)
l
Pl (cos  )
(11.8a)
CHAPTER 11: MODEL FORMULATION FOR ELECTRON INTERACTIONS IN a-Se
120
with the phase shifts:


( B)
l
 2k  V (r )( jl (kr)) 2 r 2 dr
(11.8b)
0
where jl (kr) are spherical Bessel functions. Salvat et al (1985) assume that:
f ( )  f ( B ) ( )   F ( ,  l ,  l )
l


( B)
(11.9a)
correction
with
F ( ,  l( B ) ,  l ) 
1
(2l  1)(sin 2 l  2 l( B )  i (1  cos 2 l )) Pl (cos  )
2k
(11.9b)
If an analytical screened Coulomb potential is assumed of the form:
Van (r )  

Z
Ae  a1r  (1  A)e  a2 r
r

(11.10)
where A, α1 and α2 are constants that characterize the material, then equation (11.8a)
becomes:
 A
1 A 

( )  2Z  2
 2
2
a2  q 2 
 a1  q
(11.11)

 a12

 a 22



 AQl  2  1  (1  A)Ql  2  1

 2k

 2k

(11.12)
f an
( B)
and the phase shifts become:

(B)
l , an
Z

k
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 f ( )  f ( B ) ( ) . Using this
assumption equation (11.6a) can be written as:
2
2
 A
d
1 A 
 
 f an( B ) ( )  2Z  2
 2
2
d
a 2  q 2 
 a1  q
 A
d
1 A 

 2 sin  2Z  2
 2
2
d
a 2  q 2 
 a1  q
2
 
Taking into account that q  2k sin   and that for the non-relativistic case k 
2
we get:
(11.13)
2mE

CHAPTER 11: MODEL FORMULATION FOR ELECTRON INTERACTIONS IN a-Se
121
2






d
A
1 A
 2 sin  2Z 


2
2 
d

 
  
 2 
2
a 2   2k sin    
 a1   2k sin  2  
 2  




2






d
A
1 A
 
 2 sin  2Z 

2
2
d
 2
 2mE    


2mE    
sin   
a 22  4
sin    
 a1  4


2



 2   








2
d
8Z
A
1 A


( S .I .)  2 sin 



8m
8m
d
ao
2
2 
2
2 
 a1  2 E sin   a 2  2 E sin   


2
 2

with a o 
2
(11.14)
2
and A = 0.4836, α1 =8.7824, α2 =1.6967 for a-Se (Salvat et al 1987). The
me 2
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:
 el  
d el
d
d
d   el d  el2 dq 2
d
d
dq
(11.15)
 
Using equation (11.11), q  2k sin   and dΩ=2πsinθdθ we calculate that:
2
d
4Z 2

dq 2 a o2 k 2
 A
1 A
 2
 2
2
a2  q 2
 a1  q



2
(11.16)
2

 
Since q   2k sin    when θ=0, q2=0 and when θ=π, q2=4k2. Thus equation (11.15) is
 2 

written as:
2
 el 
4k 2

0
4Z 2
a o2 k 2
 A
1 A
 2
 2
2
a2  q 2
 a1  q
2

 dq 2

(11.17)
CHAPTER 11: MODEL FORMULATION FOR ELECTRON INTERACTIONS IN a-Se
122
Calculating the integral we find that:
 el 
4Z 2

ao2 k 2
  A2
(1  A) 2 2(1  A) A  a12  4k 2  A 2 (1  A) 2 2(1  A) A a12  (11.18)

 2
 2
 2
ln  2
 2 
 2
ln 2 
2
2 
a 2  4k 2
a1  a 22
a 22
a1  a 22
a2 
 a 2  4k  a1
 a1  4k
with k 
2mE
.

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:
CHAPTER 11: MODEL FORMULATION FOR ELECTRON INTERACTIONS IN a-Se
df i (W )
mc

 ph,i (Z ,W )
dW
2 2 e 2 
123
(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:
df i (Q,W )  df i (W )

F (W ; Q,W )dW   Z r  (W  Q)(W  Bi )
dW
dW 
Bi
(11.20a)
with
F (W ; Q,W )   (W  W )(W   Q)   (W  Q)(Q  W )
(11.20b)

df i (W )
dW 

d
W
Bi
Zr  Zi  
(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:
inner,i


 1
d inel
( E ) 2e 4 
df i (W )
 df i (W ) 1  W  




ln

Z

dW   2 F (  ) ( E , W )(W  Bi )(Wmax  Bi ) (11.21a)
i

2 




dW
dW
W
Q
d
W
mu 
W

  
W



Q  ( E  E  W ) 2
where
(11.21b)
2
F
()
W
 W 
( E ,W )  1  
 
E W
 E W 
(11.21c)
Consequently the steps that must be followed to calculate the differential cross section are:
i. Calculation of
df i (W )
from (11.19).
dW
ii. Setting the number of electrons in the ith subshell, Zi.

iii. Calculation of the integral
df i (W )
dW  by making a fit to the data of photoelectric
dW 
W

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
distortion of the projectile wave function by the electrostatic field of the target atom. To
CHAPTER 11: MODEL FORMULATION FOR ELECTRON INTERACTIONS IN a-Se
124
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:

inner,i
inel

( E  Bi ) / 2

Bi
d ( E  2 Bi )
dW
dW
(11.22)
The Coulomb correction reduces the differential cross section near the threshold Bi 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):
q
1 dq   1 
 ( E,W ) 
Im

E q q
 (q,W ) 
with q   2

(11.23)

E  E  W . This expression for q  assumes that the energy-momentum
transfer relation for the electron moving in the medium is the same as that for a free
CHAPTER 11: MODEL FORMULATION FOR ELECTRON INTERACTIONS IN a-Se
125
electron in vacuum. The extension of the energy-loss function to q>0 from the optical
limit is made through:
 1  
W   1 
q2


Im

d
W
Im

(
W

(
W

))
 
 (0,W ) 
W
2
 (q,W )  0


(11.24)
The energy loss sum rule is:


1

 dWW Im (q,W )   2
2
no Z
(11.25)
0
with no being the density of atoms or molecules in the medium with Z electrons per atom
or molecule. The quantity W  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:

 1 
1
 ( E, w) 
dW W  Im
 F ( E,W ,W )


2E 0

(
0
,
W
)


with

 
q2
q2 
W    W  W   W   
2
2 
 
F ( E , W , W )  
W (W  W )
(11.26)
(11.27)
Equation (11.26) can be rewritten including exchange effects and indistinguishability as:

 1 
1
 exc ( E , w) 
dW W  Im 


2E 0
 (0,W ) 
exchange
indistiguishability



  (11.28)
 

1/ 2 
  F ( E , W ,W )  F ( E , W , E  W   W )  F ( E ,W , W ) F ( E , W , E  W   W ) 




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
produced by a different energy transfer from the primary electron to the struck electron.
Figure 11.1 illustrates this situation schematically.
CHAPTER 11: MODEL FORMULATION FOR ELECTRON INTERACTIONS IN a-Se
~
E2- W
E1-W
E1
126
E2
δ-ray
Eδ=E1-W
δ-ray
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):

 1  
W   1  
q 2

Im
Im 
   dW 
 W  W  
W
2m
 (q, W )  0
 (0, W )  

   dBf ( B) 

2q2


  
  E    B 

2m
 0 E


where f(B) is the OOS, E  W is the energy loss and B  W  is the previously
mentioned binding energy. The physical meaning of B, that is of W  , can be understood
from the following equation:
E  B 
where the term
2q2
2m
(11.30)
 2q 2
is the kinetic energy of a free and initially at rest electron that
2m
acquires momentum q. The region of integration on ( E  , B) plane is formed from the
following constrains:
E 
1
( E  B ) (Energy conservation)
2
E 
1
E  B  E ( E  2 B) (Momentum conservation)
2


(11.31)
B0
Therefore the differential cross section for energy loss E  in inelastic scattering with an
outer shell is derived from equations (11.28), (11.29) and (11.31):
2 E  E  2 E ( E  E  )
d
  ( E , E )   
dBf ( B)G ( E , E , B)
max( 0 , 2 E   E )
dE 
(11.32a)

  (11.29)


CHAPTER 11: MODEL FORMULATION FOR ELECTRON INTERACTIONS IN a-Se

with
G ( E , E , B) 
127
e 4
E
(4o ) 2
(11.32b)
1
1
1


E ( E   B) ( E  E )( E  B  E )
E ( E   B)( E  E )( E  B  E )
(11.32c)
The inelastic scattering cross section for energy loss E  is:
outer
 inel
(E) 
3E
4
d
 dE  dE   
E
2
outer
inel
( E )    dBf ( B) P( E, B)
0
(11.33a)
0
Where
P( E , B) 

1  ( E  B  S )( E  B  S ) 
2
 S  EB
 
ln 
F  sin 1 
 (11.33b)
,
B  ( E  B  S )( E  B  S )  E  B 
E B E  B
S  E( E  2B)
(11.33c)
sin 
and F is the incomplete elliptic integral of first kind F(φ,k)=

0
dx
1
(1  x )(1  k 2 x 2 )
2
.
CHAPTER 12: GENERAL DISCUSSION, CONCLUSIONS & FUTURE WORK
CHAPTER 12
GENERAL DISCUSSION,
CONCLUSIONS & FUTURE WORK
128
CHAPTER 12: GENERAL DISCUSSION, CONCLUSIONS & FUTURE WORK
129
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
CHAPTER 12: GENERAL DISCUSSION, CONCLUSIONS & FUTURE WORK
130
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
CHAPTER 12: GENERAL DISCUSSION, CONCLUSIONS & FUTURE WORK
131
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
CHAPTER 12: GENERAL DISCUSSION, CONCLUSIONS & FUTURE WORK
132
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.
CHAPTER 12: GENERAL DISCUSSION, CONCLUSIONS & FUTURE WORK
133
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 gainamplification. 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
CHAPTER 12: GENERAL DISCUSSION, CONCLUSIONS & FUTURE WORK
134
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
CHAPTER 12: GENERAL DISCUSSION, CONCLUSIONS & FUTURE WORK
135
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.
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ABSTRACT
144
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
ABSTRACT
145
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
ABSTRACT
146
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.
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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
ΕΚΤΕΝΗΣ ΠΕΡΙΛΗΨΗ
μήτρας
αποδεικνύονται
148
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
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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
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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
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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.
ΕΚΤΕΝΗΣ ΠΕΡΙΛΗΨΗ
Τα
πρωτογενή
152
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 CosterKronig 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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153
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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154
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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155
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.