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



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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 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:

µ § (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:

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.

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.

It is not a relativistic one.

Ashley uses experimental OOSs and accounts for exchange effects.

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 µ §(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 µ § 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. µ §=m=e=1):

µ § (11.23)

with µ §. This expression for µ §assumes that the energy-momentum transfer relation for the electron moving in the medium is the same as that for a free electron in vacuum. The extension of the energy-loss function to q>0 from the optical limit is made through:

µ § (11.24)

The energy loss sum rule is:

µ § (11.25)

with no being the density of atoms or molecules in the medium with Z electrons per atom or molecule. The quantity µ § 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:

µ § (11.26)

with µ § (11.27)


Equation (11.26) can be rewritten including exchange effects and indistinguishability as:

µ §(11.28)


The exchange effects concern spin interactions. The indistinguishability can be understood as follows: an energy transfer W by the primary electron reducing its energy to E-W gives an electron which cannot be distinguished from the secondary electron of energy E-W produced by a different energy transfer from the primary electron to the struck electron. Figure 11.1 illustrates this situation schematically.

Figure 11.1. Schematical illustration of the indistinguishability between a scattered projectile with energy E1-W and a secondary electron (ä-ray) with the same energy.


The Ashley’s approximation (11.24) can be rewritten as follows (in S.I):
(11.29)

µ §


where f(B) is the OOS, µ §is the energy loss and µ § is the previously mentioned binding energy. The physical meaning of B, that is of µ §, can be understood from the following equation:

µ § (11.30)

where the term µ § is the kinetic energy of a free and initially at rest electron that acquires momentum q. The region of integration on (µ §, B) plane is formed from the following constrains:

µ §(Energy conservation)

µ §(Momentum conservation) (11.31)

µ §
Therefore the differential cross section for energy loss µ § in inelastic scattering with an outer shell is derived from equations (11.28), (11.29) and (11.31):


µ § (11.32a)
with µ § (11.32b)

(11.32c)


µ §
The inelastic scattering cross section for energy loss µ § is:

µ §µ §µ § (11.33a)

Where

µ §(11.33b)



µ § (11.33c)

and F is the incomplete elliptic integral of first kind F(ö,k)= µ §.





Chapter 12

General Discussion, Conclusions & Future Work

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 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 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 ö=0 and ö=ð. On the other hand, they have the minimum probability to be ejected at ö=ð/2 and 3ð/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 ö=0, ð and 2ð. 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 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 ìm from detector’s surface. PbO has the minimum bulk space in which electrons can be produced (a radius R=200 ìm and a depth Dmax=320 ìm) whereas CdTe has the maximum one (R=500 ìm and Dmax=660 ìm).

At the stage of primary signal formation and for the typical detector thicknesses (300-1000 ìm), 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 Eµ §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 Eµ §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 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 ìm). 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 è>ð/2 with the most probable polar angle being è=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 (Wµ §~6 eV), they could be the best choice for high signal gain-amplification. On the other hand, a-Se, a-As2Se3 and Ge have the best intrinsic spatial resolution and the minimum azimuthal uniformity. Given that at the presence of an applied electric field small azimuthal uniformity means degradation of the spatial resolution mainly at one dimension, these materials could be the best choice for high spatial resolution. Due to the fact that PbO has the minimum depth at which primary electrons are produced (Dmax=320 ìm), 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:

They have high x-ray sensitivity (Wµ §~4.5 eV) and average number of primary electrons.

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. Íevertheless, 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 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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