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 distributions have minima at cosè=-1 and 1 that is the primary electrons have the minimum probability to be ejected parallel to the incident beam’s axis either forwards or backwards. Furthermore, the number of electrons in the positive cosine values is higher than the number in the negative values which means that the electrons prefer to be emitted Figure 8.9. The polar distributions (cosè) of primary electrons for GaAs at (a) 12 keV, (b) 20 keV, (c) 30 keV and (d) 40 keV. In GaAs at Eµ §12 keV the distributions are additionally influenced by the atomic deexcitation. forwards. Actually, they prefer to be emitted at a particular polar angle è 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 cosè 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=sinè=µ §), this background is due to the Auger and CK electrons emitted at the point of x-ray incidence as well as due to the primary electrons produced by the absorption of fluorescent photons. Figure 8.10. The polar distribution (cosè) 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.
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 ìm µ § 2 ìm. 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).
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: µ § (8.1) with á1, 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 á1= (1.836µ §0.001)µ § 107, b1= (1.108 µ §53.660)µ § 10-4 ìm and c1= (0.8756µ §0.0024) ìm. Provided that for a Gaussian distribution c1=µ §ó, with ó being the standard deviation, the full width at half maximum (FWHM) at each energy was calculated as 2.35ó. Furthermore, the energy dependence of the PSF was also studied in terms of the horizontal and vertical logarithmic profile histograms at the point of x-ray incidence that provide additional information concerning the primary electron production that takes place away from the centre of the xy distributions. The photoelectric absorption of incident photons, followed by the atomic deexcitation that produces Auger and CK electrons, occurs almost exclusively at the point of x-ray incidence. In addition, incident photons that are Compton scattered also create primary electrons at the spot of x-ray incidence. Consequently, the majority (approximately 80%) of primary electrons are produced at the point of x-ray incidence. This is seen in figure 8.11 in which the majority of electrons is produced at the central pixel. Therefore the FWHM is affected only from the number of electrons created in the first neighbours of the central pixel. When the number of electrons in the first neighbours increases, the FWHM increases as well. Similarly, when the number of electrons decreases, so does the FWHM. As characteristic examples, the FWHM of the fitted PSFs as a function of incident x-ray energy E for GaSe, CdTe, PbI2 and CdZnTe is given in figures 8.12(a), 8.12(b), 8.12(c) and 8.12(d), respectively. The estimated FWHM values correspond to the current resolution limit of 2 ìm whereas for all materials and incident x-ray energies the estimation error is of the order of 10-3 ìm. The FWHM in materials like GaSe (a-Se, a-As2Se3, GaAs and Ge) at energies smaller than the K edges (for example at Eµ §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
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 ìm whereas the estimation error is of the order of 10-3 ìm. 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 Table 8.1: The average values of the FWHM for the various materials in ascending order in Zone A (Pb BLI = 15.861 keV Zone AZone BZone CMaterialFWHM (ìm)MaterialFWHM (ìm)MaterialFWHM (ìm)a-Se1.346a-Se1.345a-Se1.345a-As2Se31.372a-As2Se31.372a-As2Se31.372Ge1.404Ge1.404Ge1.403CdTe1.411CdTe1.424TlBr1.453TlBr1.455Cd0.8Zn0.2Te1.437PbO1.474Cd0.8Zn0.2Te1.470CdZnTe1.444PbI21.475PbI21.472TlBr1.455HgI21.477PbO1.476PbI21.471GaSe1.486HgI21.477HgI21.476CdTe1.497GaSe1.485PbO1.476CdZnTe1.506CdZnTe1.492GaSe1.486ZnTe1.507GaAs1.507GaAs1.509GaAs1.509ZnTe1.568ZnTe1.567Cd0.8Zn0.2Te1.509not 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
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. 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 8.15. 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. A characteristic example of the yz (depth) distributions of primary electrons is shown in figure 8.15 that presents the case of CdZnTe at energies (a) 2 keV, (b) 9 keV, (c) 10 keV, (d) 26 keV, (e) 27 keV and (f) 40 keV. The distributions are explained similar to the logarithmic profiles of the xy distributions. At energies lower than Cd K edge (figures 8.15(a)-(d)), as the energy increases the electrons are created deeper inside the bulk and the distributions become wider as a result of scatter increase. At energies higher Figure 8.16. The projection of the yz (depth) distribution for CdZnTe at 40 keV. D50% denotes the depth at which the number of electrons has fallen to half of the value at the detector’s surface and Dmax the maximum depth at which primary electrons are produced.
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 ìm 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. Table 8.2: The radius of xy distributions (R), the maximum depth at which primary electrons are produced (Dmax) and the depth at which the number of electrons has fallen to half of the value at the detector’s surface (D50% ) for the various materials at 40 keV. The materials are in ascending order with respect to the values of R.
8.4.2. Polyenergetic case. The energies of the polyenergetic spectra are higher than the K edges of a-Se, a-As2Se3, GaSe, GaAs and Ge as well as the L edges of PbO and TlBr. Consequently, in these materials both the xy and yz spatial distributions are almost the same for all the incident spectra while the values of FWHM, R, Dmax and D50% are almost constant and similar to the values given in tables 8.1 and 8.2. On the other hand, the K edges of Cd, Te and I in CdTe, CdZnTe, Cd0.8Zn0.2Te, ZnTe, PbI2 and HgI2 exist at higher energies and therefore the spatial distributions depend on the incident spectrum. For example, the xy distributions of CdTe for polyenergetic spectra in which the majority of photons have energies higher than Cd K edge (i.e. the spectrum resulting from W, kVp: 40, HVL: 1.22 mm Al, filter Al: 1.02 mm) have larger radius compared to the distributions for spectra in which the majority of photons have energies smaller than Cd K edge (i.e. the spectrum resulting from Mo, kVp: 30, HVL: 0.5 mm Al, filter Mo: 0.03 mm). Table 8.3 gives the values of R, Dmax and D50% for these materials for an x-ray spectrum resulting from Mo, kVp: 20, HVL: 0.30 mm Al with no filter, which has the minimum mean energy among all the considered polyenergetic spectra. Therefore, the values of FWHM are confined within the values given in table 8.1 whereas R, Dmax and D50% range between the values given in table 8.3 (minimum values) and table 8.2 (maximum values). CdTe,
Table 8.3: The radius of xy distributions (R), the maximum depth at which primary electrons are produced (Dmax) and the depth at which the number of electrons has fallen to half of the value at the detector’s surface (D50% ) for CdTe, CdZnTe, Cd0.8Zn0.2Te, ZnTe, PbI2 and HgI2 for an x-ray spectrum resulting from Mo, kVp: 20, HVL: 0.30 mm Al with no filter. The materials are in ascending order with respect to the values of R. MaterialR (ìm)Dmax (ìm)D50% (ìm)ZnTe7511033PbI210012051HgI210012046CdTe15020051CdZnTe15018050Cd0.8Zn0.2Te15020051
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 ìm from detector’s surface. Since the typical thickness of the photoconductors is 500 ìm, 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. 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. 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 Eµ §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 Eµ §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%. 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 Eµ §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 Figure 8.18. The energy-related number distributions of fluorescent photons produced in (a) a-As2Se3 and (b) PbI2.
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. Figure 8.20. The energy-related number distributions of fluorescent photons that escape forwards and backwards in (a) TlBr and (b) CdZnTe. 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%. 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 Eµ §30 keV in materials of category A, the majority of escaping photons is fluorescent photons. For Eµ §30 keV in materials of category A, the number of escaping primary photons increases and exceeds that of escaping fluorescent photons. The summary graphs of the energy-related number distributions of escaping photons in the materials of the four categories are presented in figure 8.22. Figure 8.21. The energy-related number distributions of escaping primary and fluorescent photons in (a) GaSe and (b) PbI2.
Figure 8.22. The summary graphs of the energy-related distributions of escaping photons in materials of (a) category A, (b) category B, (c) category C and (d) category D. Figure 8.23. The energy-related number distributions of primary electrons produced in (a) a-Se and (b) CdZnTe.
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 Eµ §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. |