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
§µ §BK, the photon is absorbed by the LIII subshell (representing the L shell), ejecting a photoelectron with energy Ee= hí-BLIII
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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 µ § and the subshell ‘s’ photoelectric cross section isµ § then: µ § (7.6) If the subshell photoelectric probability is defined as: µ § (7.7) then µ §. Thus, the subshell ‘s’ that photoelectrically absorbs the photon is determined by a Monte Carlo decision, based on the probabilitiesµ §. After the subshell ‘s’ selection, a photoelectron is ejected from that subshell with energy Ee=hí-Bs for L and M shells, and with energy Ee=hí 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 è and the azimuthal angle ö, 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): µ § (7.8) where dI/dÙ is the number of photoelectrons per solid angle, A is a constant and â=õ/c where õ is the electron velocity. One of the ways to sample both è and ö 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: The azimuthal PDF normalized to a maximum of unity: h1(ö) = cos2ö (7.9) The polar PDF normalized to a maximum of unity: µ § (7.10) where h(è) is given as: µ § (7.11) and hmax(è) is the maximum value of h(è), 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).
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: Emission of a K-fluorescence photon, with probability PF=FKùK, where FK is the fraction of the photoelectric cross section contributed by K-shell electrons (obtained from Storm and Israel (1970)) and ùK is the K-fluorescent yield. 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 ùK for the various elements (except for Hg, Tl and Pb in which the K shell does not take part in the photoelectric absorption). Table 7.1. The fraction of the photoelectric cross section contributed by K-shell electrons (FK) and the K-fluorescent yield (ùK) for the various elements (except for Hg, Tl and Pb). ElementFKùKZn0.8700.466Ga0.8690.497Ge0.8670.528As0.8660.557Se0.8640.596Br0.8640.613Cd0.8410.843Te0.8360.879I0.8350.886 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 (ù), Auger (á) 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 (ùLI, ùLII, ùLIII), the Auger (aLI, aLII, aLIII) and the Coster-Kronig yields (ckLI, ckLII) of subshells LI, LII and LIII for Br, Cd, Te, I, Hg, Tl and Pb. ElementFluorescence yieldAuger yieldCoster-Kronig yieldùLIùLIIùLIIIaLIaLIIaLIIIckLIckLIIBr0.0030.0190.0190.1850.8960.9810.8120.085Cd0.0210.0590.0610.2980.7780.9390.6810.163Te0.0400.0790.0810.4380.7470.9190.5220.174I0.0430.0850.0860.4320.7400.9140.5250.175Hg0.0860.3760.3300.1460.4970.6700.7670.126Tl0.0910.3900.3410.1470.4860.6590.7620.124Pb0.0980.4040.3520.1480.4750.6480.7530.122 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 ùs, 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 occurs is determined by a Monte Carlo decision based on the probabilities µ §, where y µ §{ù, a, ck} and µ §, requiring µ §. 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.
K and L shells and the simulated atomic transitions with the corresponding energies and probabilities for the case of I. Probability of Atomic DeexcitationSimulated Atomic TransitionTransition's ProbabilityTransition's Energy (keV)FKùÊ0.740KLIII0.82028.612KMIII0.14732.294KNIII0.03333.0461- FKùÊ0.260KLILI0.07822.793KLIILIII0.46623.760KLIIILIII0.12424.055KLIMI0.02826.909KLIIIMIII0.23427.737KLIIINIII0.03728.489KMIIMIII0.02431.363KMIIINIII0.00832.171ùLI0.043LILIII0.0090.631LIMIII0.8064.313LINIII0.1865.065áLI0.432LIMIVMV0.7973.938LIMIVNV0.1984.508LINIVNV0.0055.089ckLI0.525LILIIINV1.0000.582ùLII0.085LIIMI0.0473.780LIIMIV0.9534.221áLII0.740LIIMIMII0.0382.849LIIMIVMV0.8193.602LIIMIVNV0.1424.172ckLII0.175LIILIIINIV1.0000.244ùLIII0.086LIIIMV0.8803.938LIIINV0.1204.508áLIII0.914LIIIMIVMV0.7813.307LIIINIVNV0.0164.458LIIIMIIIMIII0.2034.557 7.4.3. Energies and directions of fluorescent photons, Auger and CK electrons As it was mentioned in chapter 5, the energy of emitted fluorescent photons, Auger and CK electrons is the difference between the binding energies of the shells that are involved in the particular transition. Therefore, if the emission of a fluorescent photon involves shells i and j then the energy of the fluorescent photon (Efl;ph) is given by Efl;ph=Bi-Bj, whereas if shells i, j and k take part in the emission of an Auger or CK electron, then the energy of Auger or CK electron (EAug/CK) is given by EAug/CK=Bi-Bj-Bk. Both fluorescent photons and Auger and CK electrons are isotropically ejected from the atom. The normalized PDF is given by: µ § (7.12) and it can be considered as the product of two independent normalized PDFs (PDF(è)=1/2sinè and PDF(ö)=1/2ð) which are sampled, using the inversion method, to produce the proper values of è and ö.
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 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 8.1. Introduction Based on the method described in Chapter 7, a Monte Carlo code has been developed in order to run a set of in silico experiments using 39 monoenergetic spectra, with energies between 2 and 40 keV, and 53 mammographic spectra, in which the majority of photons have energies between 15 and 40 keV obtained from Fewell and Shuping (1978). The list of mammographic spectra is presented in table 8.1. Table 8.1. The 53 mammographic spectra used in the simulation process. The x-ray photons (107 in number) are incident at the center of a detector with dimensions 10 cm width, 10 cm length and 1 mm thickness, consisting of the already mentioned set of materials. The choice of 1 mm thickness of the photoconductors was made so that the number of both primary and fluorescent photons that escape forwards to be negligible, since primary and fluorescent photons are the major sources of primary electron production from their photoelectric absorption. The results obtained are grouped in four categories: Energy distributions of: (i) fluorescent photons, (ii) primary and fluorescent photons escaping forwards and backwards, (iii) primary electrons. Azimuthal and polar angle distributions of primary electrons. Spatial distributions of primary electrons. Arithmetics of: (i) fluorescent photons, (ii) primary and fluorescent photons escaping forwards and backwards, (iii) primary electrons.
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.
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 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 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). Figure 8.2. (a) An incident x-ray spectrum resulting from Mo, kVp: 40, HVL: 0.68 mm Al, filter Al: 0.51 mm and (b) the corresponding energy distributions of primary photons that escape forwards and backwards in CdTe. Figure 8.3. The energy distributions of fluorescent photons that escape forwards and backwards in CdTe for an incident x-ray spectrum resulting from Mo, kVp: 40, HVL: 0.68 mm Al, filter Al: 0.51 mm.
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. 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.
of photons have energies lower than the binding energies of Cd and Te K shells (26.711 keV and 31.814 keV respectively). 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 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 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.
In order to study the directions of the primary electrons produced, two histograms were plotted: that of azimuthal angle ö and that of the cosine of the polar angle è. 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(ö) =µ § (Sakellaris et al 2005), an expected fact since in the mammographic energy range the photoelectric effect dominates. Thus, the primary Figure 8.6. The azimuthal distributions of primary electrons in CdZnTe for incident monoenergetic spectra with energies between 2 and 40 keV. The distributions form groups which separate at 4, 5, 10, 27 and 32 keV. electrons have the maximum probability to be ejected at ö=0, ð and 2ð and the minimum one at ö=ð/2 and 3ð/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 Eµ §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 Eµ §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 Eµ §10 keV. Similarly, when the number of electrons decreases as the energy increases the distributions Figure 8.7. (a) The azimuthal distributions of primary electrons for a-Se, HgI2 and CdTe at 30 keV normalized at their maxima. (b) The histogram of the minima of the normalized azimuthal distributions for the various materials at 30 keV. shift downwards. The zones become wider the closer the azimuthal angles are to 0, ð and 2ð, 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 ð/2 and 3ð/2, and tend to become equal to the probability of ejection at 0, ð and 2ð. In other words, higher azimuthal uniformity means smaller tendency of electron ejection at 0, ð and 2ð. 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 Eµ §32 keV the azimuthal uniformity in ZnTe, PbI2 and HgI2 significantly increases. It was found that for the practical mammographic energies (15 keVµ §Eµ §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. Figure 8.8. The polar distributions (cosè) of primary electrons for GaAs at (a) 2 keV, (b) 5 keV, (c) 8 keV and (d) 10 keV. In GaAs at Eµ §10 keV there is not atomic deexcitation and therefore the distributions are influenced only by the photoelectric effect.
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. Directory: jspui -> bitstream bitstream -> College day annual report bitstream -> Review of coastal ecosystem management to improve the health and resilience of the Great Barrier Reef World Heritage Area bitstream -> A. gw student and alumni numbers summary 3 bitstream -> Advances in Imaging from the first x-ray Images bitstream -> Clustering Microarray Data within Amorphous Computing Paradigm and Growing Neural Gas Algorithm bitstream -> Навчальний посібник Для студентів економічних І правових спеціальностей немовних вузів Суми двнз "уабс нбу" 2014 Download 0.55 Mb. Share with your friends:
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