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

Table 8.6. The materials with the minimum and maximum number of fluorescent photons, escaping photons and primary electrons in the practical mammographic energy range (16-40 keV).

Chapter 9

A Preliminary Study On Final Signal Formation In a-Se

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:

The primary electrons drift in the vacuum.

A uniform electric field of the form µ §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/ìm which is typical for a-Se direct detectors.

The problem that must be solved is schematically illustrated in figure 9.1.

VACUUM

Figure 9.1. Schematical illustration of the primary electron drifting in the vacuum under the influence of a uniform electric field of the form E=V/l. Ui and Uf are the initial and final electron velocities in respect and F is the electric force imposed on electrons. The z-Global System is also shown.

The information concerning the initial energies Ek;i, positions (xi, yi, zi) and directions (èi, öi) 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 (èf, öf) of electrons reaching the detector’s top electrode.

The Newton’s law is:

µ § (9.1)

whereas the theorem for kinetic energy change is:

µ § (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:

µ § (9.3)

Position of primary electrons:

µ § (9.4a)

µ § (9.4b)

µ § (9.4c)

Drifting time of primary electrons:

µ § (9.5a)

µ § (9.5b)

Direction of primary electrons:

µ § (9.6)

µ §

(9.7)
9.3. Results and Discussion

As characteristic examples the results of energy, angular, spatial and time distributions of primary electrons on detector’s top electrode are presented for an x-ray spectrum resulting from Mo, kVp: 30, HVL: 0.5 mm Al, filter Mo: 0.03 mm.

9.3.1. Energy distribution of primary electrons on top electrode

Figure 9.1 presents the energy distribution of primary electrons on top electrode for an x-ray spectrum resulting from Mo, kVp: 30, HVL: 0.5 mm Al, filter Mo: 0.03 mm.

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 ìm).
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 ìm depth. The signal (electrical pulse) has a duration less than 7.2 x 10-11 s.

Figure 9.2. The time distribution (drifting time) of primary electrons on detector’s top electrode for an x-ray spectrum resulting from Mo, kVp: 30, HVL: 0.5 mm Al, filter Mo: 0.03 mm.
9.3.3. Spatial distribution of primary electrons on top electrode

FWHM of the PSF of primary electrons on top electrode is approximately 7.5 ìm, which is 5.5 times larger than the initial FWHM (1.345 ìm).

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 ìm x 3 ìm. 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 ö=0 and ð. 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 è=ð/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

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.

(a)

(c)

(b)

(d)

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.

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 è>ð/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 è~111o. Finally, figure 9.6 presents a grey scale image representing the frequency of appearance of the different (è, ö) pairs.

Figure 9.6. The grey scale image representing the frequency of appearance of the different (è, ö) pairs.

Chapter 10

Electric Field Considerations In a-Se

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 (ìhôh) is 30 x 10-6 cm2/V whereas that for electrons (ìeôe) 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 ìm, 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 ìm. As holes drift towards the active matrix, some of them land in the gaps between the pixels contributing in a loss of charge. As the number of holes in the gaps increases, the electric field is locally inverted and hence some of the trapped holes drift back inside a-Se bulk.

Figure 10.1. A schematic of the simplified cross section considered for an a-Se direct conversion digital detector to calculate the electric field distribution. The top electrode is an ITO (Indium Tin Oxide) electrode at a positive bias Vtop el=5000 V. The a-Se thickness is da-Se=500 ìm, 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 ìm. Holes land at the gaps between the pixels resulting in a surface density of positive charges ógap(x,y).

After equilibrium is achieved a constant hole concentration µ §can be considered in the gaps.

Pang et al solved the Laplace’s equation in three dimensions:

µ § (10.1)

in both the a-Se and the insulator layers with the following boundary conditions:

µ § (10.2)

µ § (10.3)

If (x,y) is on the pixel electrodes

µ § (10.4)

If (x,y) is in the gap region

µ § (10.5)

In the above equations ä denotes the infinitesimal small whereas the quantities åa-Se and åSiO2 are the dielectric constants of the a-Se and the insulator layers in respect (åa-Se=6.3, åSiO2=3.8997). In the case that ógap(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:

µ § (10.6)

The assumption is that there is no space charge in a-Se and insulator layers (except at their interface). The electric field is calculated from µ §.
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).

(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, åa-Se=å1 and åSiO2=å2):

µ §

a-Se layer:

(10.7)
µ §

Insulator layer:
(10.8)
The coefficients A00, Amn, B00 and Bmn are calculated as follows:

From equation (10.3) the relation below is obtained:

µ § (10.9)

It is defined:

µ § (10.10)

which satisfies the following equation:

µ § (10.11) The expressions for µ §and µ §depend on whether the boundary condition (10.5) (ógap(x,y) is known) or (10.6)(Ez(x,y) is known) is used. Therefore:
If ógap(x,y) is known:

(10.12)
µ §

µ § (10.13)

µ §

µ §

(10.14)

If Ez(x,y) is known:

µ § (10.15)

µ §

µ § (10.16)

(10.17)
Multiplying both sides of equation (10.11) by µ § and integrating over x and y, it is obtained:

µ § (10.18)

where

(10.20)

µ § (10.19) µ §

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 µ § in (10.18) by µ §, where Nmax is an integer. Equation (10.18) was transformed into the matrix equation:

EC=I (10.21)

where E is a (Nmax+1)2 x (Nmax+1)2 matrix and C, I are (Nmax+1)2x 1 matrices. The system (10.21) is solved using the Gauss-Jordan Elimination method (Numerical Recipies in Fortran 77, Press et al). Once Cmn is known the potential in the a-Se layer can be calculated from equations (10.7) and (10.10) as (Pang et al 1998):

µ §(10.22)
where

µ § (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 ìm, dSiO2= 5 ìm, Vp= 10 V, Tx=Ty= 50 ìm, ôx=ôy=45 ìm and Nmax=50.

Figure 10.4. A snapshot during the convergence on the solution for the potential distribution V(x, y, z) at the pixel plane (z=D) for the case that Ez(x,y)=0 in the gap region. The input parameters are: Vtop el= 5000 V, da-Se=500 ìm, dSiO2= 5 ìm, Vp= 10 V, Tx=Ty= 50 ìm, ôx=ôy=45 ìm and Nmax=50.

Chapter 11

Model Formulation For Electron Interactions In a-Se

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:

µ § (11.1)

where ëtot is the total mean free path. The number of molecules (atoms) per unit volume is:

µ § (11.2)

where NA is the Avogadro’s number, AM is the molecular weight and ñ is the density. The electrons are assumed to undergo only elastic and inelastic scattering. Thus, the total interaction cross section is defined as:

µ § (11.3)

where óel and óinel are the elastic and inelastic scattering cross sections respectively. Thus, the mean free path is defined as:

µ § (11.4)

11.3. Decision on the type of electron interaction

From equation (11.3) it is derived that µ §. If the probabilities for elastic (Pel) and inelastic (Pinel) scattering are defined as:

µ § (11.5a)

µ § (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):

µ § (11.6a)

where

µ § (11.6b)

As it is stated by Salvat et al (1985) the scattering amplitude µ § can be calculated by using the first Born approximation and some additional concepts to compensate for the fact that the Born cross section is not valid for small electron energies. Within the range of validity of the Born approximation, that is for relatively large energies of the incident electron (500 eV-50 keV), the Born (B) scattering amplitude is given by (for a measuring system with m=e=1):

µ § (11.7)

where µ §is the momentum transfer. Equation (11.7) can be written as:

µ § (11.8a)

with the phase shifts:

µ § (11.8b)

where µ § are spherical Bessel functions. Salvat et al (1985) assume that:

µ § (11.9a)

with

µ § (11.9b)

If an analytical screened Coulomb potential is assumed of the form:

µ § (11.10)

where A, á1 and á2 are constants that characterize the material, then equation (11.8a) becomes:

µ § (11.11)

and the phase shifts become:

µ § (11.12)

where Ql are the Legendre functions of the second kind. Therefore, from equation (11.6a) using equations (11.9), (11.11) and (11.12) and additional calculating ideas Salvat et al (1985) calculate the elastic scattering cross section.

For the sake of simplicity, it can be assumed that even for low electron energies the Born approximation is valid and therefore (11.9a) becomes µ §. Using this assumption equation (11.6a) can be written as:

µ §

µ § (11.13)

Taking into account that µ § and that for the non-relativistic case µ § we get:

µ §

µ §

µ § (11.14)

with µ § and A = 0.4836, á1 =8.7824, á2 =1.6967 for a-Se (Salvat et al 1987). The elastic scattering angle è of the electron can be sampled from equation (11.14) using the rejection method.

11.4.2. Elastic scattering cross section (óel)

The elastic scattering cross section is given by:

µ § (11.15)

Using equation (11.11), µ § and dÙ=2ðsinèdè we calculate that:

µ § (11.16)

Since µ §when è=0, q2=0 and when è=ð, q2=4k2. Thus equation (11.15) is written as:

µ § (11.17)
Calculating the integral we find that:

(11.18)
µ §

with µ §.
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:

Gives the subshells ionization cross sections.

Gives the energy of the ejected secondary electrons.

Assumes that the direction of the incident electron is not changed during the interaction process. Thus angular distributions are not given.

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, óph,i(Z,W), is:

µ § (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:

µ § (11.20a)

with

µ § (11.20b)

µ § (11.20c)

and È 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:

µ §

(11.21a)

where µ § (11.21b)

µ § (11.21c)

Consequently the steps that must be followed to calculate the differential cross section are:

Calculation of µ § from (11.19).

Setting the number of electrons in the ith subshell, Zi.

Calculation of the integral µ § by making a fit to the data of photoelectric cross section and integrating analytically.

Rejection method to sample the energy loss W of the incident electron.

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